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Inventiones mathematicae (2024) 237:121–220
https://doi.org/10.1007/s00222-024-01258-0
Mean curvature flow with generic initial data
Otis Chodosh1·Kyeongsu Choi2·Christos Mantoulidis3·Felix Schulze4
Received: 3 May 2022 / Accepted: 2 April 2024 / Published online: 23 April 2024
© The Author(s) 2024
Abstract
We show that the mean curvature flow of generic closed surfaces in R3avoids asymp-
totically conical and non-spherical compact singularities. We also show that the mean
curvature flow of generic closed low-entropy hypersurfaces in R4is smooth until it
disappears in a round point. The main technical ingredient is a long-time existence
and uniqueness result for ancient mean curvature flows that lie on one side of asymp-
totically conical or compact shrinking solitons.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
2Preliminaries.................................. 130
3 Linearized rescaled flow equation . . . . . . . . . . . . . . . . . . . . . . . 134
4 Dynamics of smooth ancient rescaled flows . . . . . . . . . . . . . . . . . 144
5 Uniqueness of smooth one-sided ancient rescaled flows . . . . . . . . . . . 148
6 A family of smooth ancient rescaled flows . . . . . . . . . . . . . . . . . . 153
7 Existence of a smooth ancient shrinker mean convex flow . . . . . . . . . 156
8 Long-time regularity of the flow . . . . . . . . . . . . . . . . . . . . . . . 165
F. Schulze
felix.schulze@warwick.ac.uk
O. Chodosh
ochodosh@stanford.edu
K. Choi
choiks@kias.re.kr
C. Mantoulidis
christos.mantoulidis@rice.edu
1Department of Mathematics, Bldg. 380, Stanford University, Stanford, CA 94305, USA
2School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu,
Seoul 02455, Republic of Korea
3Department of Mathematics, Rice University, Houston, TX 77005, USA
4Department of Mathematics, Zeeman Building, University of Warwick, Gibbet Hill Road,
Coventry CV7 4AL, UK
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122 O. Chodosh et al.
9 Uniqueness and regularity of one-sided ancient Brakke flows . . . . . . . 176
10 Generic mean curvature flow of low entropy hypersurfaces . . . . . . . . . 177
11 The first non-generic time for flows in R3.................. 182
Appendix A: Geometry of asymptotically conical shrinkers . . . . . . . . . . 196
Appendix B: Non-standard Schauder estimates . . . . . . . . . . . . . . . . . 198
Appendix C: Brakke flow uniqueness of regular mean curvature flows . . . . . 199
Appendix D: Ilmanen’s localized avoidance principle . . . . . . . . . . . . . . 200
Appendix E: The Ecker–Huisken maximum principle . . . . . . . . . . . . . . 204
Appendix F: Weak set flows of cones . . . . . . . . . . . . . . . . . . . . . . . 204
Appendix G: Brakke flows with small singular set . . . . . . . . . . . . . . . . 206
Appendix H: Localized topological monotonicity . . . . . . . . . . . . . . . . 212
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
1Introduction
1.1 Overview of results
Mean curvature flow is the analog of the heat equation in extrinsic differential geom-
etry. A family of surfaces M(t)⊂R3flows by mean curvature flow if
∂
∂t x⊥=HM(t)(x), (1.1)
where HM(t)(x)denotes the mean curvature vector of the surface M(t) at x. Unlike
the traditional heat equation, mean curvature flow is nonlinear. As a result, the mean
curvature flow starting at a closed surface M⊂R3is guaranteed to become singular
in finite time. There are numerous possible singularities and, in general, they can lead
to a breakdown of (partial) regularity and of well-posedness. A fundamental problem,
then, is to understand singularities as they arise.
A common theme in PDEs arising in geometry and physics is that a generic so-
lution exhibits better regularity or well-posedness behavior than the worst-case sce-
nario. This aspect of the theory of mean curvature flow has been guided by the fol-
lowing well-known conjecture of Huisken [75, #8]:
A generic mean curvature flow has only spherical and cylindrical singularities.
The implications of this conjecture on the partial regularity and well-posedness of
mean curvature flow is an important field of research in itself. See Sect. 1.2 for the
state of the art on the precise understanding of the effects of spherical and cylindrical
singularities on the partial regularity and well-posedness of mean curvature flow.
The most decisive step toward Huisken’s conjecture was taken in the trailblazing
work of Colding–Minicozzi [42], who proved that spheres and cylinders are the only
linearly stable singularity models for mean curvature flow. In particular, all remaining
singularity models are linearly unstable and ought to occur only non-generically. See
Sect. 1.3 for more discussion.
In this paper we introduce a new idea and take a second step toward the genericity
conjecture and confirm that a large class of unstable singularity models are, in fact,
avoidable by a slight perturbation of the initial data. Roughly stated, we prove:
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Mean curvature flow with generic initial data 123
The mean curvature flow of a generic closed embedded surface in R3encoun-
ters only spherical and cylindrical singularities until the first time it encoun-
ters a singularity (a) with multiplicity ≥2, or (b) that has a cylindrical end but
which is not globally a cylinder.
Cases (a) and (b) are conjectured to not occur (see the nonsqueezing conjecture and
the no cylinder conjecture in [78]). This would yield Huisken’s conjecture in full.
Using a similar method, we also prove a related statement for hypersurfaces in R4:
The mean curvature flow starting from a generic hypersurface M⊂R4with
low entropy remains smooth until it dissapears in a round point.
In particular, this gives a direct proof of the low-entropy Schoenflies conjecture (re-
cently announced by Bernstein–Wang).
Our genericity results rely on keeping simultaneous track of flows coming out of
a family of auxiliary initial surfaces on either side of M. The key ingredient is the
following new classification result of ancient solutions to mean curvature flow that
lie on one side of an asymptotically conical or compact singularity model:
For any smooth asymptotically conical or compact self-shrinker , there is a
unique ancient mean curvature flow lying on one side of √−t for all t<0.
The flow exhibits only multiplicity-one spherical or cylindrical singularities.
See Sect. 1.4 for more detailed statements of our results, and Sect. 1.5 for a discussion
of the method and the technical ingredient.
1.2 Singularities in mean curvature flow
Thanks to Huisken’s monotonicity formula, if Xis a space-time singular point of a
mean curvature flow M, it is possible to perform a parabolic rescaling around Xand
take a subsequential (weak) limit to find a tangent flow M[67,76]. A tangent flow
is always self-similar in the sense that it only flows by homotheties. If the t=−1
slice of the flow is a smooth hypersurface , then satisfies
H+1
2x⊥=0,
where His the mean curvature vector of and x⊥is the normal component of x.
In this case, we call aself-shrinker. The tangent-flow Mat a time t<0 is then
√−t, though possibly with multiplicity.
The simplest shrinkers are the generalized cylinders: Rn−k×Sk(√2k),k=
0,...,n. However, there are known to be many more examples: [1,28,81,82,87].
See also the earlier numerical work [2,41,75].
In general, non-cylindrical singularities (in the sense of generalized cylinders)
can cause a breakdown in partial regularity or well-posedness of the flow (cf.
[2,75,106]). It has thus been desirable to find situations where only cylindri-
cal singularities arise and to use this information to analyze the partial regularity
and well-posedness of the flow. To that end, Huisken classified generalized cylin-
ders as the only self-shrinkers with positive mean curvature [67,68] (and bounded
curvature, cf. [45,107]). This has led to a strong understanding of mean curva-
ture flow in the mean convex case thanks to Huisken–Sinestrari [69–71], White
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124 O. Chodosh et al.
[105,107,111], Brendle and Brendle–Huisken [19,24], Haslhofer–Kleiner [63,64],
Angenent–Daskalopoulos–Šešum [3,4], and Brendle–Choi [22,23].
The next level of difficulty is to understand flows of surfaces in R3that needn’t be
globally mean convex, but which happen to only experience multiplicity-one cylin-
drical singularities. There have been major recent advances on this topic. Colding–
Minicozzi [45] proved (using their earlier work [44], cf. [47]) that mean curvature
flows in R3having only multiplicity-one cylindrical tangent flows are completely
smooth at almost every time and any connected component of the singular set is con-
tained in a time-slice. More recently, Choi–Haslhofer–Hershkovits showed [39] (see
also [40]) that there is a (space-time) mean-convex neighborhood of any cylindri-
cal singularity. In particular, combined with [89], this settles the well-posedness of a
mean curvature flow in R3with only multiplicity-one cylindrical tangent flows.
For flows of general surfaces in R3, which may run into arbitrary singularities, our
understanding of mean curvature flow near a singular point is quite limited at present.
The most fundamental issue is the potential for higher multiplicity to arise when tak-
ing rescaled limits around a singular point. Nonetheless, some important information
is available about the tangent flows at the first singular time due to important results
of Brendle [20] classifying genus zero shrinkers in R3and of Wang [102] showing
that a smooth finite genus shrinker in R3has ends that are smoothly asymptotically
conical or cylindrical. Besides the issue of multiplicity, another problem is the huge
number of potential shrinkers that could occur as tangent flows, greatly complicat-
ing the analysis of the flow near such a singular point. (This issue presumably gets
considerably worse for hypersurfaces in Rn+1.)
1.3 Entropy and stability of shrinkers
Huisken has conjectured [75, #8] that cylinders and spheres are the only shrinkers
that arise in a generic (embedded) mean curvature flow. This conjecture provides a
promising way of avoiding the latter problem mentioned above.
Huisken’s conjecture was reinforced by the numerical observation that non-
cylindrical self-shrinkers are highly unstable. This instability was rigorously formu-
lated and proven in the foundational work of Colding–Minicozzi [42]. They defined
the entropy
λ(M) := sup
x0∈R3
t0>0M
(4πt0)−n
2e−1
4t0|x−x0|2
and observed that t→ λ(Mt)is non-increasing along any mean curvature flow, by
virtue of Huisken’s monotonicity formula. Moreover, they proved that any smooth
self-shrinker with polynomial area growth, other than generalized cylinders (i.e.,
Rn−k×Sk(√2k) with k=0,...,n), can be smoothly perturbed to have strictly
smaller entropy. This result has been used fundamentally in [7,46](cf.[88]), though
we will not need to make explicit use of it in this paper.
There have been many important applications of Colding–Minicozzi’s classifi-
cation of entropy-stable shrinkers. First, they showed their result can be used to
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Mean curvature flow with generic initial data 125
define a piecewise mean curvature flow that avoids non-spherical compact self-
shrinkers. This idea has been used to classify low-entropy shrinkers, beginning with
the work of Colding–Ilmanen–Minicozzi–White [46] who showed that the round
sphere Sn⊂Rn+1has the least entropy among all non-planar self-shrinkers. Sub-
sequently, Bernstein–Wang extended this to show that the round sphere has least en-
tropy among all closed hypersurfaces [7](seealso[113]) and that the cylinder R×S1
has second least entropy among non-planar self-shrinkers in R3[8]. Bernstein–Wang
have recently used these classification results, along with a surgery procedure, to
show that if M3⊂R4has λ(M) ≤λ(S2×R), then Mis diffeomorphic to S3[11]
(see also [9]).
1.4 Our perturbative statements
Let us describe our main perturbative results. First, we have a low-entropy result
in R4:
Theorem 1.1 Let M3⊂R4be any closed connected hypersurface with λ(M ) ≤
λ(S2×R).There exist arbitrarily small C∞graphs Mover Mso that the mean
curvature flow starting from Mis smooth until it disappears in a round point.
We state and prove this ahead of our result for R3because its statement and proof
are simpler. The low-entropy assumption allows us to perturb away all unstable sin-
gularities (in the sense of Colding–Minicozzi) and thus obtain a fully regular nearby
flow. In fact, Theorem 1.1 is a special case of Theorem 10.1, which applies in all
dimensions under suitable conditions. See also Theorem 10.7 and Corollary 10.8 for
results showing that the above behavior is generic in a precise sense.
Theorem 1.1 immediately implies the following low-entropy Schoenflies theorem,
recently announced by Bernstein–Wang (cf. [16, p. 4]).1
Corollary 1.2 (Bernstein–Wang [14]) If M3⊂R4is a closed connected hypersurface
with λ(M) ≤λ(S2×R),then Mbounds a smoothly standard 4-ball and is smoothly
isotopic to a round S3.2
For generic mean curvature flow of embedded surfaces in R3, we show more:
Theorem 1.3 Let M2⊂R3be a closed embedded surface.There exist arbitrarily
small C∞graphs Mover Mso that:
(1) the (weak)mean curvature flow of Mhas only multiplicity-one spherical and
cylindrical tangent flows until it goes extinct,or
1We emphasize that our proof of Theorem 1.1 relies heavily on several of Bernstein–Wang’s earlier works
[7,8,11] and as such our proof here of Corollary 1.2 has several features in common with their announced
strategy. The key point here, however, is that our study of generic flows in Theorem 1.1 allows us to
completely avoid the need for any surgery procedure or the refined understanding of expanders obtained
in [10,12,13,15,16].
2The isotopy from Mto the round S3follows from Theorem 1.1, and the fact that Mbounds a smooth
4-ball is then a consequence of the Isotopy Extension Theorem (cf. [65, §8, Theorem 1.3]).
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126 O. Chodosh et al.
(2) there is some T>0so that the previous statement holds for times t<T and at
time Tthere is a tangent flow of Mthat either
(a) has multiplicity ≥2, or
(b) has a cylindrical end,but is not a cylinder.
Note two things:
•In the R3theorem, unlike in the low-entropy higher dimensional theorems, we
need to make use of a weak notion of mean curvature flow because we are plac-
ing no entropy assumptions and are thus interested in flowing through spherical
and cylindrical singularities. See Theorem 11.1 for the precise statement, which
includes the notion of weak mean curvature flow that we make use of.
•Both of the potential tangent flows in case (2) are conjectured to not exist (see the
nonsqueezing conjecture and the no cylinder conjecture in [78]).
There are two features of our work that distinguish it from previous related work:
•We only need to perturb the initial condition. See [42] for a piecewise flow con-
struction that perturbs away compact singularity models (see also [98]).
•We are able to perturb away (certain) non-compact singularity models.
1.5 Our perturbative method: ancient one-sided flows
For a fixed hypersurface M0⊂Rn+1, one has a weak mean curvature flow t→M0(t )
starting at M0. Suppose that X=(x,T ) is a singular point for t→M0(t). The usual
method for analyzing the singularity structure at Xis to study the tangent flows of
t→M0(t ) at X, i.e., the (subsequential) limit of the flows
t→λ(M0(T +λ2t) −x)=: Mλ
0(t)
as λ→∞. As discussed above, by Huisken’s monotonicity formula, for t<0, this
will weakly (subsequentially) converge to a shrinking flow t→ M(t) associated to
a (weak) self-shrinker.
Our new approach to generic mean curvature flow is to embed the flow t→M0(t )
in a family of flows by first considering a local foliation {Ms}s∈(−1,1)and flowing the
entire foliation, simultaneously, by mean curvature flow t→ Ms(t ). The avoidance
principle for mean curvature flow implies that Ms(t) ∩Ms(t ) =∅for s= s.The
entire foliation can be passed to the limit simultaneously, i.e., we can consider the
flows
t→λ(Ms(T +λ2t) −x):= Mλ
s(t)
and send λ→∞.
If we choose s0 diligently as λ→∞, then after passing to a subsequence,
t→Mλ
s(t) will converge to a non-empty flow t→ ¯
M(t) that stays on one side of the
original tangent flow t→M(t) and which is ancient, i.e., it exists for all sufficiently
negative t. If we can prove that the one-sided ancient flow t→ ¯
M(t) has certain nice
properties (i.e., only cylindrical singularities), then we can exploit this to find a choice
of ssmall so that t→Ms(t ) is well behaved.
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Mean curvature flow with generic initial data 127
We proceed to give more details as to how we exploit this ancient one-sided
flow, t→ ¯
M(t). Assume that the tangent flow to M0(t) at Xis smooth and has
multiplicity one, so M(t) =√−t for t<0. Then, considering the rescaled flow
τ→eτ
2¯
M(−eτ), we note that eτ
2¯
M(−eτ)lies strictly on one side of and
=lim
τ→−∞eτ
2¯
M(−eτ)(1.2)
(a priori, this could occur with multiplicity, but in practice one can rule this out by
upper semi-continuity of density). In the current work, we will deal with all that
are: (i) compact but not spheres, or (ii) non-compact with asymptotically (smoothly)
conical structure. These tangent flows encompass all the necessary ones for our afore-
mentioned theorem statements, by virtue of L. Wang’s [102] characterization of the
asymptotic structure of non-compact singularity models.
Our definitive rigidity theorem of ancient one-sided flows is:
Theorem 1.4 Let n⊂Rn+1be a smooth self-shrinker that is either compact or
asymptotically (smoothly)conical.Up to parabolic dilation around (0,0)∈Rn+1×
R,there exists a unique3ancient solution to mean curvature flow t→ ¯
M(t) so that
¯
M(t) is disjoint from √−t and has entropy <2F().
Remark There has recently been an outburst of activity regarding the rigidity of an-
cient solutions to geometric flows. We mention here [5,21–23,25–27,48–50,61,
72,100]. In the setting at hand, Theorem 1.4 was motivated from the recent work
in [38] on the classification of compact ancient solutions of gradient flows of el-
liptic functionals in Riemannian manifolds. However, this is the first time that the
one-sidedness condition has been exploited so crucially, and geometrically, in the
setting of ancient geometric flows. In the elliptic setting, there have been interest-
ing exploitations of one-sided foliations by minimal surfaces; see, e.g., Hardt–Simon
[60], Ilmanen–White [79], and Smale [95]. Our current parabolic setting, however,
presents a number of complications that come from the fact that the shrinkers we
are interested in are primarily noncompact, and thus the flows cannot be written as
global perturbations of the self-similarly shrinking solution.
Remark Neither of the hypothesis in Theorem 1.4 can be removed. There can be
many ancient flows that intersect √−tand converge to as t→−∞after rescal-
ing; see Theorem 6.1.Also,fora≥0, the grim reaper in the slab R×(a , a +π) is
a nontrivial example of an ancient flow that is disjoint from its tangent flow at −∞,
2[R×{0}].
Next, we show that ¯
M(t) encounters only generic singularities for as long as it
exists. We establish many properties of ¯
M(t) in Theorem 9.1, and some the important
ones are summarized here.
Theorem 1.5 Let t→ ¯
M(t),n⊂Rn+1be as in Theorem 1.4 and 2≤n≤6. Then:
3For technical reasons, the long-time aspect of the existence statement currently requires 2 ≤n≤6. If one
only cares about sufficiently negative times, existence and uniqueness hold true for all dimensions.
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128 O. Chodosh et al.
•The flow t→ ¯
M(t) only has multiplicity-one,generalized cylindrical singularities:
Rn−k×Sk(√2k),k=1,...,n.
•At t=0, ¯
M(0)is smooth and star-shaped.
•If is noncompact,then t→ ¯
M(t) exists for all t∈Rand
lim
t→∞
1
√t¯
M(t)
is an outermost expander associated to the asymptotic cone of .
To prove Theorem 1.5, we show that the one-sided ancient flow t→ ¯
M(t) must
be shrinker mean convex; geometrically, this means that the rescaled flow moves
in one direction. This is where the one-sided property is crucially used. Recalling
the spectral instability of shrinkers discovered in [42], and that only the first eigen-
function of the linearization of Gaussian area along has a sign, we show that the
evolution of a one-sided flow is dominated by the first eigenfunction, which in turn
yields shrinker mean convexity. Shrinker mean convexity is preserved under the flow
and can be used analogously to mean convexity to establish regularity of the flow
(cf. [85,88,96,105,107]). We emphasize that our analysis of the flow ¯
M(t) in
Theorem 1.5 is influenced by the work of Bernstein–Wang [8] where they studied a
(nearly ancient) flow on one side of a asymptotically conical shrinker of low-entropy.
Because we do not assume that the flow has low-entropy (besides assuming the limit
at −∞ has multiplicity one), we must allow for singularities (while in [8], the flow
is a posteriori smooth). In particular, this complicates the analysis of the flow near
t=0 significantly.
Finally, we explain how Theorems 1.4 and 1.5 can be used to prove the main
results of the paper, Theorems 1.1 and 1.3. We begin by considering the setting of
Theorem 1.1, namely M3⊂R4with λ(M) ≤λ(S2×R). Up to performing an initial
perturbation using [42], we can assume this inequality is strict λ(M) < λ(S2×R).
As described above, we embed Min a local foliation {Ms}s∈(−1,1)in space with
supsλ(Ms)<λ(S2×R). We now flow the entire foliation simultaneously, obtain-
ing flows {Ms(t)}. Suppose that M0(t) encounters a singularity at (x,T).Ourlow-
entropy assumption and work of Bernstein–Wang [11] implies that any tangent flow
to M0(t) at (x,T) is associated to some compact or asymptotically (smoothly) coni-
cal self-shrinker .Ifis a round sphere, we are done. Otherwise, we may combine
λ(Ms)<λ(S2×R)with Theorems 1.4 and 1.5 to find that the Ms(t ) are free of
singularities at points that are captured by the ancient flow on one side of .
The major issue is that points y∈Ms(t) close to (x,T ) but with t−T|y−x|2
will not be captured by a one-sided flow. In this case, if we rescale Ms(t) around
(x,T) so that (y,t)is moved to a point of unit distance, and pass to limits, we instead
obtain a flow that agrees with the shrinking for t<0 and is some unknown flow
flowing out of the cone at infinity of for t>0. The insight is that, by parabolic
cone-splitting, these flows will have strictly lower Gaussian density than . As such,
the flow Ms(t) improves as compared to M0(t) in that it has a lower-density maximal
density singular point. We iterate this finitely many times to prove Theorem 1.1.
We now describe the necessary modifications to prove Theorem 1.3. Consider
M2⊂R3(without any entropy bounds). Arguing as above, we can flow a foliation
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Mean curvature flow with generic initial data 129
{Ms}s∈(−1,1)and use Theorems 1.4 and 1.5 to show that Ms(t) is well-approximated
by the ancient-one sided flow in some neighborhood of a compact or asymptotically
(smoothly) conical self-shrinking singularity of M0(t). At this point we do not use
the density drop argument described above, but instead must rely on a genus mono-
tonicity argument. To do this, we note that such a self-shrinking singularity must have
genus >0 by a result of Brendle [20]. On the other hand, the ancient one-sided flow
is star-shaped at time 0 by Theorem 1.5. Using this, we find that the one-sided flow
strictly loses genus when bypassing the singularity. Thus, after finitely many pertur-
bations there cannot be any singularities that are not round spheres or cylinders. This
proves Theorem 1.3.
Remark There have been several significant results related to this paper that appeared
between the time the paper first appeared and this version. On one hand, the results
of this paper were extended to shrinkers with asymptotically cylindrical ends in [34].
On the other hand, the density drop argument was generalized into a standalone tool
to prove generic regularity results for low-entropy flows in [32,37]. This density
drop argument was also used to generalize the Hardt–Simon generic regularity result
[60] for area-minimizing hypersurfaces from 8 to 9 and 10 ambient dimensions [36].
In terms of its use in the current paper, the first- and fourth-named authors, along
with Daniels-Holgate recently proved [35] that the outermost level set flows are com-
pletely smooth for a short time after the occurrence of a singularity modeled on an
asymptotically conical self-shrinker. In particular, this result would allow us to avoid
the density drop argument used here altogether. Finally, we mention the recent ma-
jor breakthrough by Bamler–Kleiner who proved [6] the multiplicity-one conjecture
in R3.
1.6 Other results
We list several other new results we’ve obtained in this work that might be of inde-
pendent interest:
•For any smooth compact or asymptotically conical shrinker , we construct an I
parameter family of smooth ancient mean curvature flows (where Iis the index of
as a critical point of Gaussian area, as defined in (3.8)) that—after rescaling—
limit to as t→−∞; see Theorem 6.1.
•We show that the outermost flows of the level set flow of a regular cone are smooth
self-similarly expanding solutions. We also construct associated expander mean
convex flows that converge to the given expander after rescaling; see Theorem 8.21.
•We include a proof of a localized version of the avoidance principle for weak
set flows due to Ilmanen; see Theorem D.3. This implies a strong version of the
Frankel property for shrinkers; see Corollary D.4.
•We improve known results concerning the connectivity of the regular set of a unit-
regular Brakke flow with sufficiently small singular set. See Corollary G.5.
•We localize the topological monotonicity of White [103]. In particular, our results
should be relevant in the context of the strict genus reduction conjecture of Ilmanen
[78, #13]. See Appendix Hand the proof of Proposition 11.4.
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130 O. Chodosh et al.
1.7 Organization of the paper
In Sect. 2we recall some conventions and definitions used in the paper.
The main technical work of the paper is contained in Sects. 3–9, which establish
the existence and uniqueness, together with regularity of ancient one-sided flows.
The geometric applications of this existence and uniqueness result are then given in
Sects. 10 and 11. As such, the reader less interested in the technicalities in prov-
ing existence/uniqueness of the one-sided ancient flow may want to jump straight to
Sect. 10.
More precisely, in Sect. 3we analyze the linearized graphical mean curvature flow
equation over an asymptotically conical shrinker. We use this to study the nonlinear
problem in Sect. 4. These results are applied in Sect. 5to prove our main analytic
input, Corollary 5.2, the uniqueness of ancient one-sided graphical flows.
Section 6contains a construction of the full I-parameter family of ancient flows.
This is not used elsewhere, since we construct the one-sided flows by GMT methods
allowing us to flow through singularities. We begin this GMT construction in Sect. 7
where we construct an ancient one-sided Brakke/weak-set flow pair. In Sect. 8we
establish optimal regularity of the ancient one-sided flow. We put everything together
in Sect. 9and give the full existence and uniqueness statement for the ancient one-
sided flows.
We apply this construction to the study of the mean curvature flow of generic low
entropy hypersurfaces in Sect. 10 and to the study of the first non-generic time of the
mean curvature flow of a generic surface in R3in Sect. 11.
In Appendix Awe improve some decay estimates for asymptotically conical ends
of shrinkers. In Appendix Bwe recall Knerr’s non-standard parabolic Schauder es-
timates. In Appendix Cwe prove that mean curvature flows with bounded curvature
and controlled area ratios are unique in the class of Brakke flows. We prove Ilmanen’s
localized avoidance principle in Appendix D. Appendix Erecalls the non-compact
Ecker-Huisken maximum principle. In Appendix Fwe study weak set flows coming
out of cones. We show that Brakke flows with sufficiently small singular set have
connected regular part in Appendix G. Finally, in Appendix Hwe localize certain
topological monotonicity results.
2Preliminaries
In this section we collect some useful definitions, conventions, and useful ways to
recast mean curvature flow, which we will make use of in the sequel.
2.1 Spacetime
We will often consider the spacetime of our mean curvature flows, Rn+1×R, with
its natural time-projection map t:Rn+1×R→R:
t(x,t):=t.
For any subset E⊂Rn+1×Rwe will denote
E(t) :={x∈Rn+1:(x,t)∈E}.
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Mean curvature flow with generic initial data 131
2.2 The spacetime track of a classical flow
Let us fix a compact n-manifold M, possibly with boundary. Suppose that f:M×
[a,b]→Rn+1is a continuous map that is smooth on M◦×(a, b](where M◦=
M\∂M) and injective on each M×{t}for t∈[a,b]. Assume that t→ f(M◦,t) is
flowing by mean curvature flow. Then, we call
M:={f(M,t)×{t}:t∈[a, b]}⊂Rn+1×R
aclassical mean curvature flow and define the heat boundary of Mby
∂M:=f(M,a)∪f(∂M,[a,b]).
By the maximum principle, classical flows that intersect must intersect in a point that
belongs to either one of their heat boundaries (cf. [103, Lemma 3.1]).
2.3 Weak set flows and level set flows
If ⊂Rn+1×R+(where R+=[0,∞)could be shifted as necessary) is a closed
subset of spacetime, then M⊂Rn+1×Ris a weak set flow (generated by )if:
(1) Mand coincide at t=0 and
(2) if Mis a classical flow with ∂Mdisjoint from Mand Mdisjoint from ,
then Mis disjoint from M.
We will often consider the analogous definition with R+replaced by Rin which case
one should omit requirement (1).
There may be more than one weak set flow generated by a given . See [103].
However, there is one weak set flow that contains all other weak set flows generated
by . It is called the level set flow (or biggest flow). For ⊂Rn+1×R+as above,
we define it inductively as follows. Set
W0:={(x,0):(x,0)/∈}
and then let Wk+1be the union of all classical flows Mwith Mdisjoint from
and ∂M⊂Wk. We define the level set flow (or biggest flow) generated by as:
M:=(Rn+1×R+)\(∪∞
k=0Wk)⊂Rn+1×R+.
See [58,74,111] for further references for weak set flows and level set flow.
We will sometimes engage in a slight abuse of notation, referring to a weak set
flow (or a level set flow) generated by a closed subset 0⊂Rn+1, when we really
mean that it is generated by 0×{0}(or a suitable time-translate) in the sense defined
above.
2.4 Integral Brakke flows
Another important notion of weak mean curvature flow is a Brakke flow (cf. [18,74]).
We follow here the conventions used in [112].
An (n-dimensional) integral Brakke flow in Rn+1is a 1-parameter family of Radon
measures (μ(t))t∈Iover an interval I⊂Rso that:
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132 O. Chodosh et al.
(1) For almost every t∈I, there exists an integral n-dimensional varifold V(t)with
μ(t) =μV(t) so that V(t)has locally bounded first variation and has mean cur-
vature Horthogonal to Tan(V (t ), ·)almost everywhere.
(2) For a bounded interval [t1,t
2]⊂Iand any compact set K⊂Rn+1,
t2
t1K
(1+|H|2)dμ(t )dt < ∞.
(3) If [t1,t
2]⊂Iand f∈C1
c(Rn+1×[t1,t
2])has f≥0 then
f(·,t
2)dμ(t
2)−f(·,t
1)dμ(t
1)
≤t2
t1−|H|2f+H·∇f+∂
∂t fdμ(t)dt.
We will often write Mfor a Brakke flow (μ(t))t∈I, with the understanding that we’re
referring to the family It→μ(t) of measures satisfying Brakke’s inequality.
A key fact that relates Brakke flows to weak set flows, which we will use implicitly
throughout the paper, is that the support of the spacetime track of a Brakke flow is a
weak set flow [74, 10.5].4
2.5 Density and Huisken’s monotonicity
For X0:=(x0,t
0)∈Rn+1×Rconsider the (backward) heat kernel based at (x0,t
0):
ρX0(x,t):=(4π(t0−t))−n
2exp −|x−x0|2
4(t0−t),(2.1)
for x∈Rn+1,t<t
0. For a Brakke flow Mand r>0weset
M(X0,r):=x∈Rn+1ρX0(x,t
0−r2)dμ(t
0−r2). (2.2)
This is the density ratio at X0at a fixed scale r>0. Huisken’s monotonicity formula
[67](cf.[76]) implies that
d
dt ρX0(x,t)dμ(t)≤−H−(x−x0)⊥
2(t −t0)
2
ρX0(x,t)dμ(t)
so in particular, we can define the density of Mat X0by
M(X0):= lim
r0M(X0,r). (2.3)
4The definition of Brakke flow used in [74] is slightly different than the one given here, but it is easy to
see that the proof of [74, 10.5] applies to our definition as well.
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Mean curvature flow with generic initial data 133
2.6 Unit-regular and cyclic Brakke flows
An integral Brakke flow M=(μ(t))t∈Iis said to be
•unit-regular if Mis smooth in some space-time neighborhood of any spacetime
point Xfor which M(X) =1;
•cyclic if, for a.e. t∈I,μ(t) =μV(t) for an integral varifold V(t) whose unique
associated rectifiable mod-2 flat chain [V(t)]has ∂[V(t)]=0(see[109]).
Integral Brakke flows constructed by Ilmanen’s elliptic regularization approach [74]
(see also [112, Theorem 22]) are unit-regular and cyclic. More generally, if Miare
unit-regular (resp. cyclic) integral Brakke flows with MiM, then Mis also unit-
regular (resp. cyclic) by [108](cf.[92, Theorem 4.2]; resp. [109, Theorem 4.2]).
Recall that a sequence of integral Brakke flows Miconverges to an integral Brakke
flow M, denoted MiM,if
(1) μi(t) μ(t ) for each t, and
(2) for a.e. t, we can pass to a subsequence depending on tso that Vi(t ) V (t ) as
varifolds.
The motivation for this definition of convergence is that these are the conditions that
follow (after passing to a subsequence) if we have local mass bounds for Miand
seek to prove a compactness theorem (cf. [74, §7]).
2.7 Shrinkers
A smooth hypersurface ⊂Rn+1is a self-shrinker if
H+1
2x⊥=0,(2.4)
where His the mean curvature vector of and x⊥is the normal component of x.
We will always assume that has empty boundary, unless specified otherwise. One
can easily check that (2.4) is equivalent to any of the following properties:
•t→√−tis a mean curvature flow for t<0,
•is a minimal hypersurface for the metric e−1
2n|x|2gRn+1,or
•is a critical point of the F-functional
F():=(4π)−n
2
e−1
4|x|2
among compactly supported deformations, as well as translations and dilations.
See [42, §3].
We will say that is asymptotically conical there is a regular cone C(i.e., the cone
over a smooth submanifold of Sn) so that λ →Cin C∞
loc(Rn+1\{0})as λ0.
Remark By considering the t0 limit (in the Brakke flow sense) of the flow t→
√−t, we see that limλ0λ is unique in the Hausdorff sense, so the asymptotic
cone of must be unique. Moreover, because we have assumed that the convergence
is in C∞
loc, there is no potential higher multiplicity in the limit (see, e.g., [102, §5]).
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134 O. Chodosh et al.
2.8 Curvature conventions
Consider ⊂Rn+1open with ∂ =a self-shrinker. Write νfor the unit normal
vector field to that points into . We define the second fundamental form 2-tensor
Aat each p∈to equal
A:Tp×Tp→R,A
(ξ , ζ ) =−Dξν·ζ.
Recall that dual to Ais the shape operator or Weingarten map, defined at each
p∈to be the tangent space endomorphism given by
S:Tp→Tp, S(ξ) =−Dξν.
We fix the sign of the scalar mean curvature Has follows
H=Hν.
Thus, H=trA, with the principal curvatures of being the eigenvalues of A.
With these conventions, the shrinker mean curvature from (2.4) can be written as
H+1
2x⊥=H+1
2x·νν.
For example, the sphere bounding a unit ball has normal vector pointing to the in-
side, positive mean curvature, and positive principal curvatures. Conversely, the same
sphere bounding the complement of a closed unit ball has normal vector pointing to
the outside, negative mean curvature, and negative principal curvatures.
2.9 Entropy
Following [42], one uses the backward heat kernel ρ(x0,t0)from (2.1) to define the
entropy of a Radon measure μon Rn+1by
λ(μ) := sup
x0∈Rn+1
t0>0ρ(x0,t0)(x,0)dμ. (2.5)
Then, one can define the entropy of an arbitrary Brakke flow M=(μ(t))t∈Iby:
λ(M):=sup
t∈I
λ(μ(t)). (2.6)
Huisken’s monotonicity formula implies that t→λ(μ(t)) is non-increasing.
3 Linearized rescaled flow equation
Let n⊂Rn+1be a smooth properly immersed asymptotically conical shrinker.5
5The analysis here also holds in the much simpler case of compact .
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Mean curvature flow with generic initial data 135
3.1 Spectral theory in Gaussian L2space
We consider the following operator on :
Lu :=u−1
2x·∇u+1
2u+|A|2u. (3.1)
This is the “stability” operator for the F-functional in Sect. 2.7 in the sense that
d2
ds2s=0F(graph(su)) =−u(Lu) ρ ,
for any compactly supported function u:→R, where ρis the Gaussian weight
ρ(x):=(4π)−n
2e−1
4|x|2,(3.2)
i.e., ρ:= ρ(0,0)(·,−1)in the notation of (2.1). See [42, Theorem 4.1]. This stability
operator, (3.1), is only self-adjoint if we work on Sobolev spaces weighted by ρ.We
thus define a weighted L2dot product for measurable functions u,v:→R:
u, vW:= u, v ρdHn.(3.3)
This induces a metric ·Wand a Hilbert space
L2
W() := {u:→R:uW<∞}.(3.4)
Likewise, we define the higher order weighted Sobolev spaces
Hk
W() := {u:→R:uW+∇uW+···+∇k
uW<∞}.(3.5)
They are Hilbert spaces for the dot product
u, vW,k :=u, vW+∇u, ∇vW+···+∇k
u, ∇k
vW,(3.6)
whose induced norm is denoted ·W,k. It is with respect to these weighted measures
spaces that Lis self-adjoint, i.e.,
Lu, vW=u, LvW,∀u, v ∈H2
W(). (3.7)
We have:
Lemma 3.1 There exist real numbers λ1≤λ2≤... and a corresponding complete
L2
W-orthonormal set ϕ1,ϕ
2,...:→Rsuch that Lϕi=−λiϕiand limiλi=∞.
Proof This follows from the standard min-max construction of eigenvalues and eigen-
functions and the compactness of the inclusion H1
W() ⊂L2(), in the spirit of the
Rellich–Kondrachov theorem, proven in [8, Proposition B.2].
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136 O. Chodosh et al.
Since λj→∞as j→∞, there exist I,K∈Nsuch that
λ1≤···≤λI<0≤λI+1=0=···=λI+K<λ
I+K+1≤.... (3.8)
For notational convenience,for any binary relation ∼∈{=,=,<,>,≤,≥} we define
the spectral projector ∼μ:L2
W() →L2
W() given by:
∼μ:f→
j:λj∼μf, ϕjWϕj.(3.9)
We wish to study solutions of the inhomogeneous linear PDE
(∂
∂τ −L)u =hon ×R−,(3.10)
where R−=(−∞,0]in all that follows. (Of course, in practice, hmay depend on u.)
At a formal level, if u(·,τ)∈H2
W(·,τ) and h(·,τ)∈L2
W() for τ∈R−, then we
can use Lemma 3.1 and Hilbert space theory to decompose
u(·,τ)=: ∞
j=1
uj(τ )ϕj,h(·,τ)=: ∞
j=1
hj(τ )ϕj,(3.11)
where the uj,hj:R−→Rare expected (by virtue of (3.10)) to be solutions of
u
j(τ ) =−λjuj(τ ) +hj(τ ). (3.12)
Turning this formal argument into a rigorous one is standard:
Lemma 3.2 (Weighted L2estimate) Fix δ>0, 0 <δ
<min{δ,−λI}.Suppose that
0
−∞ e−δτ h(·,τ)W2
dτ < ∞.(3.13)
There exists a unique solution u(“strong in L2”)of (3.10)such that
<0(u(·,0)) =0,(3.14)
0
−∞ e−δτ(u(·,τ)W,2+∂
∂τ u(·,τ)W)2
dτ < ∞.(3.15)
It is given by the series representation in (3.11)with coefficients:
uj(τ ) := −0
τ
eλj(σ −τ)hj(σ ) dσ, j =1,...,I, (3.16)
uj(τ ) := τ
−∞ eλj(σ −τ)hj(σ ) dσ, j =I+1,I +2,.... (3.17)
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Mean curvature flow with generic initial data 137
Moreover,for every τ∈R−,
e−δτu(·,τ)W≤C0
−∞ e−δσ h(·,σ)W2
dσ1
2,(3.18)
where C=C(δ, δ,λ
1,...,λ
I).
Proof The proof is a straightforward computation and adaptation of Galerkin’s
method from linear parabolic PDE. One starts with “weak L2” solutions ([57, §7.1.2,
Theorems 3, 4]) and upgrades them to strong ones ([57, §7.1.3, Theorem 5]).
3.2 Weighted Hölder space notation
Let ⊂. We assume that the injectivity radius of at points in is at least i0>0.
For k∈N,α∈(0,1), we will use the following notation for the standard Cknorm,
Cαseminorm, and Ck,α norm:
fk;:=
k
i=0
sup
|∇i
f|,(3.19)
[∇k
f]α;:= sup
x=y∈
d(x,y)<i0
|∇k
f(x)−Py→x∇k
f(y)|
d(x,y)α(3.20)
for Py→xparallel transport defined along the unique minimizing geodesic from y
to x,
fk,α;:=fk;+[∇k
f]α;.(3.21)
Now let d∈R. We define the weighted counterparts of the quantities above:
f(d)
k;:=
k
i=0
sup
x∈˜r(x)−d+i|∇i
f(x)|,(3.22)
[∇k
f](d)
α;:= sup
x=y∈
d(x,y)<i0
1
˜r(x)d−α+˜r(y)d−α|∇k
f(x)−Py→x∇k
f(y)|
d(x,y)α,(3.23)
f(d)
k,α;:=f(d)
k;+[∇k
f](d−k)
α;.(3.24)
Above, ˜ris as in [31], so we briefly remind the reader what it is. Recall from [31,
Sect. 2] that [31, Lemma 2.3] gives a diffeomorphism C\BR(0)×[R,∞)on the
non-compact part of , where is the link of the asymptotic cone C. We will thus
parametrize points of by (ω, r ) ∈×[R,∞). We emphasize that the coordinate
ralong is not exactly equal to dRn+1(·,0)(like it is along the cone). Then ris
extended to ˜rdefined on all of so that ˜r≥1onand ˜r=routside of BRfor
R≥1 as above.
In any of the above estimates, if we don’t indicate the domain over which the
norm is taken, then it must be understood to be =.
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138 O. Chodosh et al.
3.3 Pointwise estimates
We fix δ0∈(0,−λI)and α∈(0,1)throughout the section.
We revisit the inhomogeneous linear PDE
(∂
∂τ −L)u =hon ×R−.(3.25)
We will treat classical solutions of the PDE, i.e., ones that satisfy it pointwise. We
use implicitly throughout the fact that regularity on hyields improved regularity on
uby standard (local) parabolic Schauder theory.
Lemma 3.3 (Interior C2,α estimate) Suppose u,hsatisfy (3.25),
sup
τ∈R−
e−2δ0τh(·,τ)(−1)
0,α <∞,(3.26)
<0(u(·,0)) ≡0,(3.27)
and
0
−∞ e−δ0τ(u(·,τ)W,2+∂
∂t u(·,τ)W)2
dτ < ∞.(3.28)
Then,for every τ∈R−and compact K⊂Rn+1,
e−δ0τu(·,τ)2,α;∩K≤Csup
σ∈R−
e−2δ0σh(·,σ)(−1)
0,α ,(3.29)
with C=C(,α,δ0,K).
Proof Lemma 3.2 applies with δ∈(δ0,2δ0)and δ=δ0<min{δ,−λI}by virtue of
(3.26), (3.27), (3.28), and gives
e−δ0τu(·,τ)W≤c0
−∞ e−δσ h(·,σ)W2
dσ1
2.
Apply the non-standard Schauder estimate in Corollary B.2 of Appendix Bon ∩K,
where Kis a compact set containing Kin its interior. It shows that, for τ≤0:
u(·,τ)C2,α (∩K)
≤C(,α,K)uL1((∩K)×[τ−1,τ ])+sup
σ∈[τ−1,τ ]h(·,σ)0,α;∩K
≤C(,α,δ0,K)
eδ0τ0
−∞ e−δσ h(·,σ)W2
dσ1
2
+sup
σ∈[τ−1,τ ]h(·,σ)0,α;∩K
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Mean curvature flow with generic initial data 139
≤C(,α,δ0,K)
eδ0τ0
−∞(e−δσ h(·,σ)(−1)
0)2dσ1
2
+sup
σ∈[τ−1,τ ]h(·,σ)(−1)
0,α
≤C(,α,δ0,K)
eδ0τ·sup
σ∈R−
e−2δ0σh(·,σ)(−1)
0+e2δ0τ
·sup
σ∈R−
e−2δ0σh(·,σ)(−1)
0,α .
This gives (3.29).
Lemma 3.4 (Global C0estimate) Suppose u,hsatisfy (3.25). If
lim
τ→−∞u(·,τ)W=0,(3.30)
and
τ
τ−1(u(·,σ)2
W,2+∂
∂t u(·,σ)2
W)dσ <∞,(3.31)
for all τ∈R−,then for all τ∈R−and R≥R0():
u(·,τ)(1)
0≤Csup
σ≤τu(·,σ)(1)
0;∩BR(0)+h(·,σ)(0)
0;\BR(0),(3.32)
for C=C().
Proof Fix τ0∈R−. Following [31, Lemma 3.15], we consider ϕ:=α|x|−βwith
α:=2sup
(\BR(0))×(−∞,τ0]|h|+2R−1sup
(∩∂BR(0))×(−∞,τ0]|u|,
β:=4sup
(\BR(0))×(−∞,τ0]|h|+R−1sup
(∩∂BR(0))×(−∞,τ0]|u|.
Note that, by definition,
∂
∂τ ϕ≡0,(3.33)
and, if R≥2,
ϕ>0on\BR(0). (3.34)
Consider the function
f:=u−ϕon ( \BR(0)) ×(−∞,τ
0].
As in [31, Lemma 3.15], by construction:
f≤0on( ∩∂BR(0)) ×(−∞,τ
0],(3.35)
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140 O. Chodosh et al.
(∂
∂τ −L)f ≤0on( \BR(0)) ×(−∞,τ
0](3.36)
for Rsufficiently large. Multiply (3.36)byf+ρ, where f+:= max{f, 0}and ρis as in
(3.2), and integrate over \BR(0).Using(3.7), and differentiating under the integral
sign using (3.31), (3.33), and [57, §5.9.2, Theorem 3] we have, for a.e. τ≤τ0:
\BR(0)|∇f+|2ρdHn
≤\BR(0)
(1
2f2
++|A|2f2
+)ρ dHn−1
2\BR(0)∂
∂τ f2
+ρHn
≤(1
2+O(R−2)) \BR(0)
f2
+ρdHn−1
2∂
∂τ \BR(0)
f2
+ρdHn.
Plugging into Ecker’s Sobolev inequality [52](cf.[31, Proposition 3.9]) we get:
(R2−4n−8+O(R−2)) \BR(0)
f2
+ρdHn≤−8∂
∂τ \BR(0)
f2
+ρdHn.(3.37)
We take R0≥2 large enough so that the above computation holds and R2−4n−8+
O(R−2)>0 whenever R≥R0.By(3.30), (3.34),
lim
τ→−∞\BR(0)
f2
+ρdHn=0.
Because [57, §5.9.2, Theorem 3] shows absolute continuity of \BR(0)f2
+ρdHn
with respect to τ≤τ0, integrating (3.37) over (−∞,τ
0], we find that f+≡0on
\BR(0)×(−∞,τ
0]. Therefore, f≤0on×(−∞,τ
0]. Thus, on \BR(0),
˜r−1u(·,τ
0)≤2sup
(\BR(0))×(−∞,τ0]|h|+2R−1sup
(∩∂BR(0))×(−∞,τ0]|u|.
Redoing this with −fin place of fimplies (3.32).
Lemma 3.5 (Global C2,α estimate) Suppose u,hsatisfy (3.25). Then,
u(·,τ)(1)
2,α ≤Csup
σ≤τu(·,σ)(1)
0+h(·,σ)(−1)
0,α ,(3.38)
for all τ∈R−,with C=C(,α).
Proof It suffices to prove
u(·,0)(1)
2,α ≤Csup
τ∈R−u(·,τ)(1)
0+h(·,τ)(−1)
0,α ,
since the general claim will follow by translation in time.
Define the function :×(−∞,0)→Rn+1so that t→ (x,t) tracks the
normal movement in Rn+1of x∈by mean curvature:
∂
∂t (x,t) =Ht((x, t )),
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Mean curvature flow with generic initial data 141
where t:=√−t, and (·,−1)≡Id. Note that the map
t(x):→,(x,t)→ 1
√−t(·,t)
satisfies
∂
∂t t=1
√−t(·,t)
(−2t) +Ht((·,t))
=1
√−t
T(·,t)
(−2t) =1
(−2t)T
t.(3.39)
Let τ0∈R−be arbitrary. Set:
u(·,τ):= u(·,τ +τ0),
h(·,τ):= h(·,τ +τ0),
v(·,t):=√−tut(·), −log(−t).
Note that v(·,t)makes sense for t∈(−∞,−eτ0]⊃(−∞,−1]. Noting that (3.39)is
the same evolution equation as considered in [31, Definition 3.7], we find that, as in
[31, (3.3)], the transformed function ˆvsatisfies
∂
∂tv(x,t)=tv(x,t)+|At|2v(x,t)+1
√−t
ht(x), −log(−t)(3.40)
as a function on t.
We have, for R=R() sufficiently large and t∈[−e, −eτ0],
v(·,t)0;t∩(B4R(0)\BR(0)) ≤Cu(·,−log(−t))(1)
0.(3.41)
Likewise,
1
√−t
ht(·), −log(−t)0,α ;t∩(B4R(0)\BR(0)) ≤C
h(·,−log(−t))(−1)
0,α ,(3.42)
with C=C(,α,R).
By Knerr’s Schauder estimates (Theorem B.1 in Appendix B) applied to suffi-
ciently small balls, and (3.41), (3.42), we find that for R=R() sufficiently large
v(·,−eτ0)2,α;−eτ0∩(B3R(0)\B2R(0))
≤Csup
t∈[−1,−eτ0]v(·,t)0;t∩(B4R(0)\BR(0))
+ 1
√−t
ht(·), −log(−t)0,α;t∩(B4R(0)\BR(0))
≤Csup
t∈[−1,−eτ0]u(·,−log(−t))(1)
0+
h(·,−log(−t))(−1)
0,α
≤Csup
τ∈R−u(·,τ)(1)
0+h(·,τ)(−1)
0,α .
Undoing the renormalization for ˆv, we thus find
u(·,0)(1)
2,α;∩B3Re−τ0/2(0)\B2Re−τ0/2(0)≤Csup
τ∈R−u(·,τ)(1)
0+h(·,τ)(−1)
0,α .
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142 O. Chodosh et al.
Taking the supremum over all τ0∈R−,wehave
u(·,0)(1)
2,α;\B2R(0)≤Csup
τ∈R−u(·,τ)(1)
0+h(·,τ)(−1)
0,α .
Along with standard interior parabolic Schauder estimates, this yields (3.38).
3.4 Nonlinear error term
We work in graphical coordinates over .Onitself, we denote the position vector
by x, and we fix a unit normal νso that following our conventions from Sect. 2.8
we can form the mean curvature scalar H=H·ν. For graphical surfaces S=
graphu, with unit normal ν(so that ν·ν>0) and mean curvature Hthe rescaled
mean curvature flow is:
∂
∂τ u=vH+1
2x·ν,(3.43)
where vis the geometric function
v:=(1+|(Id +uA)−1(∇u)|2)1
2=(ν ·ν)−1.(3.44)
We can rewrite (3.43)as
(∂
∂τ −L)u =E(u) on ×R−,(3.45)
where we take Lto be precisely the operator from (3.1) and
E(u) := vH+1
2x·ν−H+1
2x·ν−Lu. (3.46)
Note that the second term in parentheses vanishes, since satisfies the shrinker equa-
tion, but it is helpful to keep this vanishing term in mind in terms of estimating the
error. The nonlinear error term can be estimated as follows:
Lemma 3.6 There exists η=η() such that for u:→Rwith u(1)
2≤η,the
nonlinear error term E(u) from (3.46)decomposes as
E(u)(x)=u(x)E1(x,u(x), ∇u(x), ∇2
u(x))
+∇u(x)·E2(x,u(x), ∇u(x), ∇2
u(x)), (3.47)
where E1,E2are smooth functions on the following domains:
E1(x,·,·,·):R×Tx×Sym(Tx⊗Tx) →R,
E2(x,·,·,·):R×Tx×Sym(Tx⊗Tx) →Tx.
Moreover,we can estimate:
˜r(x)2+j−|∇i
x∇j
z∇k
q∇
AE1(x,z,q,A)|
≤C(˜r(x)−1|z|+|q|+r(x)|A|)max{0,1−j−k−},(3.48)
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Mean curvature flow with generic initial data 143
˜r(x)1+j−|∇i
x∇j
z∇k
q∇
AE2(x,z,q,A)|
≤C(˜r(x)−1|z|+|q|+r(x)|A|)max{0,1−j−k−}.(3.49)
In the above,C=C(),˜ris as in Sect.3.2,and i,j,k,≥0.
Proof It will be convenient to rewrite (3.46)as
E(u) =vH −H−u−|A|2u
=:EH(u)
+1
2v(x·ν) −x·ν+x·∇u−u
=:Ex·ν(u)
.
By linearity, it suffices to check (3.47), (3.48), (3.49) separately for EH(u),Ex·ν(u).
Using [31, (C.4)], EH(u) readily decomposes as
EH(u) =uEH
1(·,u,∇u, ∇2
u) +∇u·EH
2(·,u,∇u, ∇2
u).
Estimates (3.48), (3.49)forEH
1,EH
2are a simple consequence of scaling; indeed,
they are the scale-invariant manifestation of the quadratic error nature of the lineariza-
tion of Hon an asymptotically conical manifold where, crucially, |A|+˜r|∇A|≤
C˜r−1.
Using [31, (C.2)], the term Ex·ν(u) can in fact be written as
Ex·ν(u) =∞
k=1
ukAk
(x,∇u), (3.50)
where Ak
is the 2-tensor corresponding to the k-times composition of the shape
operator (the dual to A). Note that this is also of the required form, (3.47), and in fact
it can be viewed as both uEx·ν
1or ∇u·Ex·ν
2. The power series in (3.50) is absolutely
convergent by [31, Lemma 2.7]. By the sharp derivative estimate in Corollary A.3 of
Appendix A, the series in (3.50) can also be differentiated and estimated termwise to
yield (3.48)ifweviewitasuEx·ν
1,or(3.49)ifweviewitas∇u·Ex·ν
2.
Corollary 3.7 There exists η=η() such that for u:→Rwith u(1)
2≤η:
˜r|E(u)|≤C(˜r−1|u|+|∇u|)(˜r−1|u|+|∇u|+˜r|∇2
u|), (3.51)
E(u)(−1)
0,α ≤Cu(1)
1,αu(1)
2,α,(3.52)
and for ¯u:→Ralso with ¯u(1)
2≤η:
˜r|E(¯u) −E(u)|≤C(˜r−1|u|+|∇u|+˜r|∇2
u|+˜r−1|¯u|+|∇¯u|+˜r|∇2
¯u|)
·(˜r−1|¯u−u|+|∇(¯u−u)|+˜r|∇2
(¯u−u)|), (3.53)
E(¯u) −E(u)(−1)
0,α ≤C(u(1)
2,α +¯u(1)
2,α)¯u−u(1)
2,α.(3.54)
Above,C=C(),and (3.51), (3.53)are pointwise estimates on .
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144 O. Chodosh et al.
Proof Estimates (3.51), (3.52) follow by applying (3.47) to decompose E(u) and
(3.48), (3.49) with i=j=k==0 to estimate the two terms in the decomposition.
Estimates (3.53), (3.54) follow by applying (3.47) to decompose E(u),E(¯u),us-
ing the fundamental theorem of calculus to expand
E1(·,¯u, ∇¯u, ∇2
¯u) −E1(·,u,∇u, ∇2
u),
E2(·,¯u, ∇¯u, ∇2
¯u) −E2(·,u,∇u, ∇2
u),
and then using (3.48), (3.49) with i=0, j+k+=1 to estimate the Taylor expan-
sion coming from the fundamental theorem of calculus.
4 Dynamics of smooth ancient rescaled flows
In what follows, we make extensive use of the L2projection notation from (3.9).
Lemma 4.1 Suppose u,hare such that
(∂
∂τ −L)u =h, (4.1)
and that for some μ∈{λ1,...,λ
I}∪{0}:
lim
τ→−∞eμτ >μ u(·,τ)W=0.(4.2)
Suppose that hsatisfies,respectively for each binary relation >,=,<,that
|h(·,τ),
μu(·,τ)W|≤δ(τ)u(·,τ)Wμu(·,τ)W(4.3)
for some non-decreasing δ:R−→[0,δ
0].If δ0is sufficiently small depending on ,
then
>μu(·,τ)W≤Cδ(τ)≤μu(·,τ)W,∀τ∈R−,(4.4)
and either
=μu(·,τ)W≤Cδ(τ)<μu(·,τ)W,∀τ∈R−,(4.5)
or there exists a non-decreasing τ0:R−→R−such that τ0(τ ) ≤τfor all τ∈R−
and
<μu(·,τ)W≤Cδ( ¯τ)=μu(·,τ)W,∀¯τ∈R−,τ≤τ0(¯τ). (4.6)
Here,C=C().
Proof Let μ(resp. μ) be the largest (resp. smallest) eigenvalue of Lstrictly below
(resp. strictly above) μ—if μ=λ1, the choice of μis irrelevant. Taking the dot
product of (4.1) with eigenfunctions of Lwe find, by (4.3), that:
d
dτ <μu(·,τ)W+μ<μ u(·,τ)W≥−Cδ(τ)u(·,τ)W,
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Mean curvature flow with generic initial data 145
|d
dτ =μu(·,τ)W+μ=μu(·,τ)W|≤Cδ(τ)u(·,τ)W,
d
dτ >μu(·,τ)W+μ>μ u(·,τ)W≤Cδ(τ)u(·,τ)W,
for C=C(). Note that we may multiply through with eμτ and rewrite these as:
d
dτ (eμτ <μu(·,τ)W)+(μ−μ)(eμτ <μ u(·,τ)W)≥−Cδ (τ )(eμτ u(·,τ)W),
(4.7)
|d
dτ (eμτ =μu(·,τ)W)|≤C δ(τ )(eμτ u(·,τ)W), (4.8)
d
dτ (eμτ >μu(·,τ)W)+(μ −μ)(eμτ >μu(·,τ)W)≤Cδ (τ )(eμτ u(·,τ)W);
(4.9)
By the Merle–Zaag ODE lemma (see [38, Lemma B.1]), applied to (4.7), (4.8), (4.9),
together with the a priori assumption (4.2), it follows that if δ0is sufficiently small,
then
eμτ >μu(·,τ)W≤C δ(τ )(e μτ ≤μu(·,τ)W), ∀τ∈R−,
and that either6
eμτ =μu(·,τ)W≤Cδ (τ )(eμτ <μu(·,τ)W), ∀τ∈R−,
or there exists a non-decreasing τ0:R−→R−such that τ0(τ ) ≤τfor all τ∈R−
and
eμτ <μu(·,τ)W≤Cδ( ¯τ )(eμτ =μu(·,τ)W), ∀¯τ∈R−,τ≤τ0(¯τ).
This is the required result after canceling out eμτ from all sides.
Corollary 4.2 Suppose u,hare such that (4.1), (4.3)hold for all μ∈{λ1,...,λ
I}∪
{0}.If
δ(τ) ≤C0sup
σ≤τu(·,σ)W,∀τ∈R−,(4.10)
and
lim
τ→−∞u(·,τ)W=0,(4.11)
then either u≡0or there exists μ∈{λ1,...,λ
I}∪{0}and a non-decreasing τ0:
R−→R−with τ0(τ ) ≤τfor all τ∈R−such that
=μu(·,τ)W≤Cδ(¯τ)=μu(·,τ)W,∀¯τ∈R−,τ≤τ0(¯τ), (4.12)
6The Merle–Zaag ODE lemma is for a fixed coefficient δ, rather than a variable coefficient δ(·), on the right
hand sides of the differential inequalities. Per the lemma, for any fixed value δ(τ ), we have a dichotomy:
looking backwards in time, either the first alternative (“unstable dominates neutral”) holds immediately,or
the second alternative (“neutral dominates unstable”) holds eventually. Note that if either alternative holds
for one coefficient, then it must hold for all smaller coefficients (in view of the alternative) and all previous
times (by the monotonicity of δ(·)). The function τ0(·)succinctly arranges for the unavoidable fact that
the second alternative may go into effect at different times for different values of δ(·).
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146 O. Chodosh et al.
for C=C(,C0),and,if μ<0, then
0<lim inf
τ→−∞eμτ u(·,τ)W≤lim sup
τ→−∞ eμτ u(·,τ)W<∞.(4.13)
If K=0(recall,K=dim ker Lin L2
W()), then μ= 0.
Proof Let μ∈{λ1,...,λ
I}∪{0}be the smallest possible choice for which (4.2) holds
true; note that this statement isn’t vacuous, since (4.11) guarantees (4.2) at least for
μ=0.
Claim (4.5)cannot hold.
Proof of claim Note that if (4.5) held, then μ= λ1.Ifμis the largest eigenvalue
smaller than μ,by(4.1), (4.3), (4.4), and (4.5), if it did hold, we would have that
d
dτ <μu(·,τ)W+μ<μ u(·,τ)W≥−Cδ(τ)<μu(·,τ)W.
Arguing as in [38, Claim 4.5], which requires the knowledge that δ(τ) is bounded per
(4.10), it would follow that
<μu(·,τ)W≤Ce−μτ ,
at which point (4.4), (4.5) guarantee that
>μu(·,τ)W=≥μu(·,τ)W≤Cδ(τ)<μu(·,τ)W≤Ce−2μτ ,
violating the minimal nature of μ. Thus, (4.5) cannot hold.
So, (4.6) must hold. Together, (4.4), (4.6)give(4.12). If μ=0, there is nothing
left to prove; the result follows. Otherwise, we simply note that (4.1), (4.3), (4.4),
(4.6)give:
|d
dτ =μu(·,τ)W+μ=μu(·,τ)W|≤Cδ(τ)=μu(·,τ)W.(4.14)
Arguing as in [38, Claim 4.5] again, with μin place of λI, gives the rightmost in-
equality of (4.13), and the leftmost inequality is obtained by instead using the two-
sided nature of the bound in (4.14).
The following lemma verifies that assumptions (4.3), (4.10) are met for ancient
rescaled mean curvature flows that stay sufficiently close to in the suitable scale-
invariant sense:
Lemma 4.3 If u:×R−→Ris such that (3.45)and
lim
τ→−∞u(·,τ)(1)
3=0,(4.15)
then the choice
δ(τ) := sup
σ≤τu(·,σ)(1)
2,α (4.16)
satisfies (4.3)with h=E(u),and (4.10).
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Mean curvature flow with generic initial data 147
Proof First let’s show that δ(τ) satisfies (4.3) with h=E(u). We use Lemma 3.6’s
decomposition, (3.47). By virtue of (3.48) and (4.16), we only need to check that
∇u(·,τ)·E2(·,u,∇u, ∇2
u), μu(·,τ)W≤Cδ(τ)u(·,τ)Wμu(·,τ)W.
(4.17)
We deal with the cases <,=differently than >.
We can deal with <and =at the same time, and we use the symbol to denote
either of these binary relations. Since there are only finitely many eigenvalues ≤μ
by (3.8), one easily sees that:
∇μu(·,τ)W≤Cμu(·,τ)W,(4.18)
where Cdepends on ,μ. In particular, (4.18) implies (4.17)forafter integrating
by parts and using (3.49) with i+j+k+≤1.
We now deal with the binary relation >. Since
u(·,τ)=>μ u(·,τ)+=μu(·,τ)+<μu(·,τ),
we can rewrite the left hand side of (4.17)as
∇u(·,τ)·E2(·,u,∇u, ∇2
u), >μu(·,τ)W
=∇>μu(·,τ)·E2(·,u,∇u, ∇2
u), >μu(·,τ)W
+∇=μu(·,τ)·E2(·,u,∇u, ∇2
u), >μu(·,τ)W
+∇<μ u(·,τ)·E2(·,u,∇u, ∇2
u), >μu(·,τ)W
=1
2E2(·,u,∇u, ∇2
u), ∇(>μu(·,τ))
2W
+∇=μu(·,τ)·E2(·,u,∇u, ∇2
u), >μu(·,τ)W
+∇<μ u(·,τ)·E2(·,u,∇u, ∇2
u), >μu(·,τ)W.
The second and third terms we estimate via (4.18) and then μu(·,τ)W≤
u(·,τ)Wand (3.49) with i+j+k+=0.Thefirsttermweestimatebyin-
tegrating by parts and then using >μu(·,τ)W≤u(·,τ)Wand (3.49) with
i+j+k+=1. This completes our proof of (4.17) and thus (4.3) with h=E(u).
Now we check that δ(τ) satisfies (4.10). Fix R>0. By Lemma 3.5, then
Lemma 3.4, and then Corollary 3.7:
u(·,τ)(1)
2,α ≤Csup
σ≤τu(·,σ)(1)
0+E(u)(·,σ)(−1)
0,α
≤Csup
σ≤τu(·,σ)0;∩BR(0)+E(u)(·,σ)(−1)
0,α
≤Csup
σ≤τu(·,σ)0;∩BR(0)+δ(σ)u(·,σ)(1)
2,α.
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148 O. Chodosh et al.
In particular, since δ(·)=o(1)by (4.15), we deduce
δ(τ) =sup
σ≤τu(·,σ)(1)
2,α ≤Csup
σ≤τu(·,σ)(1)
0;∩BR(0).
In the compact set ∩BR(0), we can thus control the C0norm of u(·,σ) by the
L2( ∩B2R(0)×[σ−1,σ])norm of u, which is dominated by the L2
W() norm.
Thus, δ(τ) satisfies (4.10), completing the proof.
5 Uniqueness of smooth one-sided ancient rescaled flows
In this section, we characterize smooth ancient flows lying on one side of an asymp-
totically conical shrinker , with Gaussian density no larger than twice that of the
entropy of .
Lemma 5.1 (One-sided decay) Let (S(τ ))τ≤0be an ancient rescaled mean curva-
ture flow lying on one side of and such that,for τ≤0, we can write S(τ ) :=
graphu(·,τ),u≥0, with
lim
τ→−∞u(·,τ)(1)
3=0.(5.1)
Then,either u≡0, or there exists a nonzero constant α1∈Rsuch that:
lim
τ→−∞eλ1τ=λ1u(·,τ)=α1ϕ1,(5.2)
lim sup
τ→−∞ e2λ1τ=λ1u(·,τ)−α1e−λ1τϕ1W<∞,(5.3)
lim sup
τ→−∞ e2λ1τu(·,τ)−=λ1u(·,τ)W<∞,(5.4)
Proof Lemma 4.3 and (5.1) imply that Lemma 4.1, Corollary 4.2 are applicable with
δ(τ) := sup
σ≤τu(·,σ)(1)
2,α.
Invoke Corollary 4.2.Ifu≡0, there is nothing left to prove. Let us suppose u≡0.
Claim μ=λ1.
Proof of claim Note that
0≤u(·,τ)==μu(·,τ)+=μu(·,τ) =⇒ (=μu(·,τ))
−≤|=μu(·,τ)|.
By (4.12),
(=μu(·,τ))
−W≤=μu(·,τ)W
≤Cδ(¯τ)=μu(·,τ)W,∀¯τ∈R−,τ≤τ0(¯τ). (5.5)
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Mean curvature flow with generic initial data 149
Denote h(τ ) := =μu(·,τ)−1
W=μu(·,τ). Since λ1≤μ≤0, it follows from the
Rellich–Kondrachov theorem on L2
W() that h(τ ) converges after passing to a sub-
sequence to some μ-eigenfunction h(−∞)with h(−∞)W=1. By (5.5) and the fact
that limτ→−∞ δ(τ) =0, it follows that h(−∞)≥0, and the claim follows from ele-
mentary elliptic theory.
In view of μ=λ1,(4.13) implies
lim sup
τ→−∞ eλ1τδ(τ) < ∞.(5.6)
Thus,
d
dτ =λ1u(·,τ)+λ1=λ1u(·,τ)W≤Cδ(τ)=λ1u(·,τ)W
can be integrated to yield the existence of a limit limτ→−∞ eλ1τ=λ1u(·,τ), i.e.,
(5.2), and by (5.6)alsogives(5.3). Finally, we note that Lemma 4.1 is applicable
with μ=λ1. Indeed, (4.3) always holds by Lemma 4.3, while (4.2) holds by (4.12),
(5.6). Therefore, conclusion (4.4)ofLemma4.1 implies
u(·,τ)−=λ1u(·,τ)W=>λ1u(·,τ)W≤Cδ(τ)=λ1u(·,τ)W≤Ce−2λ1τ,
(5.7)
which implies (5.4).
Corollary 5.2 (One-sided uniqueness for graphical flows) Up to time translation,
there is at most one non-steady ancient rescaled mean curvature flow (S (τ ))τ≤0on
one side of satisfying (5.1).
Proof We assume that u,¯u≡ 0 are two such solutions. It follows from Lemma 5.1
that we can translate either uor ¯uin time so that
lim
τ→−∞eλ1τ(¯u−u)(·,τ)W=0.(5.8)
It will also be convenient to write δ(τ ),¯
δ(τ) for the quantities corresponding to (4.16)
for u,¯u. By Lemmas 4.3 and 5.1,
δ(τ) +¯
δ(τ) ≤C1e−λ1τ,τ∈R−(5.9)
for a fixed C1. Finally, we introduce the notation
w:= ¯u−u, Ew:= E( ¯u) −E(u),
so that
(∂
∂τ −L)w =Ew.(5.10)
Using (3.47) and the fundamental theorem of calculus,
Ew=¯uE1(·,¯u, ∇¯u, ∇2
¯u) +∇¯u·E2(·,¯u, ∇¯u, ∇2
¯u)
−uE1(·,u,∇u, ∇2
u) −∇u·E2(·,u,∇u, ∇2
u)
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150 O. Chodosh et al.
=wE1(·,u,∇u, ∇2
u)
+∇w·E2(·,u,∇u, ∇2
u)
+¯uE1(·,¯u, ∇¯u, ∇2
¯u) −E1(·,u,∇u, ∇2
u)
+∇¯u·E2(·,¯u, ∇¯u, ∇2
¯u) −E2(·,u,∇u, ∇2
u)
=wE1(·,u,∇u, ∇2
u)
+∇w·E2(·,u,∇u, ∇2
u)
+¯u1
0DzE1(···)dtw
+¯u1
0DqE1(···)dt·∇w
+¯u1
0DAE1(···)dt·∇2
w
+∇¯u·1
0DzE2(···)dtw
+∇¯u·1
0DqE2(···)dt·∇w
+∇¯u·1
0DAE2(···)dt·∇2
w, (5.11)
where, in all six instances, ··· stands for (·,u+tw,∇u+t∇w, ∇2
u+t∇2
w).
We note that we can formally write
Ew=wF +∇w·F+∇2
w·F
with
|F|+|F|+|F|≤C(δ(τ) +¯
δ(τ)).
We take the L2
Wdot product of (5.11) with wand integrate the w∇2
w·Fterms by
parts so that, in every term, we have at least two instances of wand ∇w. In partic-
ular, we will pick up derivatives of DAE1and DAE2. Furthermore, when integrating
by parts, we pick up terms of the form
w(xT⊗∇w) ·FρdHn.
Recall that Ecker’s Sobolev inequality [52](cf.[31, Proposition 3.9]) implies that
|x|f2
W≤4nf2
W,1,
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Mean curvature flow with generic initial data 151
and we can thus estimate
w(xT⊗∇w) ·FρdHn≤C(δ(τ) +¯
δ(τ ))(|x|wW∇wW)
≤C(δ(τ ) +¯
δ(τ))w2
W,1.
Using Lemma 3.6,(5.1), and (5.9), we find
|w(·,τ),Ew(·,τ)W|≤C2e−λ1τw(·,τ)2
W,1,τ∈R−,(5.12)
for a fixed C2. Here, ·W,1is the norm induced from (3.6) with k=1.
We use (5.12) to derive two estimates on the evolution of w2
W. First, together
with (5.10) and (3.8), it implies
1
2d
dτ w(·,τ)2
W=w(·,τ),Lw(·,τ)+Ew(·,τ)W
≤−λ1w(·,τ)2
W+C2e−λ1τw(·,τ)2
W,1,τ∈R−,
which in turn implies
d
dτ (e2λ1τw(·,τ)2
W)≤C2eλ1τw(·,τ)2
W,1,τ∈R−.(5.13)
Second, recalling the definition of Lin (3.1), integrating by parts, and using (5.12),
it follows that there exists a sufficiently negative τ0such that:
1
2d
dτ w2
W=−∇w2
W+w,(1
2+|A|2)w +EwW
≤−1
2∇w2
W+C3w2
W,τ≤τ0,(5.14)
withafixedC3.
We next compute the evolution of ∇w2
W. To that end, we need a couple of
preliminary computations. By the Gauss equation,
Ric(∇w,∇w) =HA(∇w, ∇w) −A2
(∇w,∇w), (5.15)
where A2
is the 2-tensor corresponding to the self-composition of the shape operator
(the dual to A). From the definition of the second fundamental form and the shrinker
equation (2.4), H+1
2x·ν=0, we have
∇(x·∇w) ·∇w=|∇w|2−2HA(∇w, ∇w) +x·∇2
w(∇w,·). (5.16)
In what follows, we recall the Gaussian density ρ, defined in (3.2), which satisfies
∇ρ=−1
2ρx. An integration by parts, followed by the Bochner formula ∇w=
∇w+Ric(∇w,·),(5.15), (5.16), implies:
(w−1
2x·∇w)2ρdHn
=
(w−1
2x·∇w)div(ρ∇w) d Hn
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152 O. Chodosh et al.
=−∇(w−1
2x·∇w) ·∇wρdHn
=−
(∇w−Ric(∇w,·)−1
2∇(x·∇w)) ·∇wρdHn
=−
(∇w−1
2x·∇2
w+A2
(∇w,·)−1
2∇w) ·∇wρdHn
=−div(ρ∇2
w) +−A2
(∇w,·)+1
2∇wρ·∇wdHn
=
(|∇2
w|2−A2
(∇w,∇w) +1
2|∇w|2)ρdHn.(5.17)
We can now estimate the evolution of ∇w2
.Using(5.10) and the definition of L
in (3.1):
1
2d
dτ ∇w2
W=∇w, ∇∂
∂τ wW
=−w−1
2x·∇w, ∂
∂τ wW
=−w−1
2x·∇w,w−1
2x·∇w+|A|2w+1
2w+EwW
=−w−1
2x·∇w2
W+∇w, ∇(|A|2w+1
2w)W
−w−1
2x·∇w,EwW
=−w−1
2x·∇w2
W+(1
2+|A|2)1
2∇w2
W
+∇w, w∇|A|2W−w−1
2x·∇w,EwW.
We claim that this implies:
1
2d
dτ ∇w2
W≤C4w2
W,1,τ≤τ0,(5.18)
with fixed C4, after possibly choosing a more negative τ0. Indeed, in the immediately
preceding expression, we use Cauchy–Schwarz on the last term, which together with
the first term yield
−w−1
2x·∇w2
W−w−1
2x·∇w,EwW
≤−1
2w−1
2x·∇w2
W+1
2Ew2
W.
In the right hand side, the −1
2w−1
2x·∇
w2
Wterm is used, via (5.17), to
dominate all ∇2
wterms in Ew, which we computed in (5.11); note that these terms
have small coefficients for sufficiently negative τby virtue of (5.9). This yields (5.18).
Together, (5.14), (5.18) imply that there exist C5≥1, C6such that
d
dτ (∇w2
W+C5w2
W)≤C6w2
W,τ≤τ0.(5.19)
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Mean curvature flow with generic initial data 153
Integrating (5.19) from −∞ to τand using the decay of w, we deduce:
w(·,τ)2
W,1≤∇w(·,τ)2
W+C5w(·,τ)2
W≤C6τ
−∞ w(·,s)2
Wds, τ ≤τ0.
(5.20)
By (5.8), we may take τ0more negative yet so that
w(·,τ)2
W≤e−2λ1τ,τ≤τ0.(5.21)
Thus, by evaluating the integral in (5.20) using the crude estimate in (5.21), we find
w(·,τ)2
W,1≤C6
2|λ1|e−2λ1τ,τ≤τ0,(5.22)
with the same τ0. Integrating (5.13) from −∞to τ, and using (5.8)at−∞ and (5.22),
we get the following improvement over (5.21):
w(·,τ)2
W≤C2C6
2|λ1|2e−3λ1τ,τ≤τ0,(5.23)
with the same τ0. Now we iterate. Using (5.20) again, with (5.23) in place of (5.21):
w(·,τ)2
W,1≤C2C2
6
3!|λ1|3e−3λ1τ,τ≤τ0,(5.24)
with the same τ0. Integrating (5.13) from −∞ to τ, and using (5.24) rather than
(5.22), we get the following improvement over (5.23):
w(·,τ)2
W≤C2
2C2
6
2·3!|λ1|4e−4λ1τ,τ≤τ0,(5.25)
with the same τ0. Repeating this k∈Ntimes altogether (we showed steps k=1, 2),
we find
w(·,τ)2
W≤Ck
2Ck
6
k!(k +1)!|λ1|2ke−(2+k)λ1τ,τ≤τ0,(5.26)
with the same τ0. Fixing τ≤τ0and sending k→∞,(5.26)givesw(·,τ)≡0.
6 A family of smooth ancient rescaled flows
In this section we construct an I-dimensional family (recall, Iis as in (3.8)) of
smooth ancient rescaled mean curvature flows that flow out of the fixed asymptot-
ically conical shrinker n⊂Rn+1as τ→−∞. Using the tools at our disposal, this
is a straightforward adaptation of [38, Sect. 3]. For the convenience of the reader, we
emphasize that this section is not used elsewhere in the paper and may be skipped on
first read. It is purely of independent interest.
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154 O. Chodosh et al.
Remark When is asymptotically conical, it seems nontrivial to verify that the con-
struction in this section proves the existence of a one-sided flow without performing
further error-term analysis near infinity. (If is compact this follows easily.) We find
it easier to instead prove this existence of one-sided flows in Sect. 7using geometric
measure theory, which we also use to show that the one-sided flow can be continued
through singularities, which is crucial for subsequent applications. We emphasize that
the uniqueness of one-sided flows was established in Sect. 5.
6.1 The nonlinear contraction
We continue to fix δ0∈(0,−λI),α∈(0,1). It will be convenient to also consider the
operator
ι−:a=(a1,...,a
I)∈RI→
I
j=1
aje−λjτϕj.(6.1)
Theorem 6.1 There exists μ0=μ0(,α,δ
0)such that,for every μ≥μ0,there exists
a corresponding ε=ε(,α, δ0,μ) with the following property:
For an y a∈Bε(0)⊂RIthere exists a unique S(a):×R−→Rso that the
hypersurfaces S(τ) := graphS(a)(·,τ) satisfy the rescaled mean curvature flow
∂
∂τ x=HS(τ)(x)+1
2x⊥,∀x∈S(τ), (6.2)
with the a priori decay
sup
τ∈R−
e−δ0τ(S(a)−ι−(a))(·,τ)(1)
2,α ≤μ|a|2(6.3)
and the terminal condition <0(S(a))(·,0)=ι−(a)(·,0).
Proof The geometric PDE (6.2) is equivalent to (3.45). Consider the affine space
C[a]:={u:×R−→R:<0(u(·,0)) =ι−(a)(·,0), u∗<∞},
where
u∗:= sup
τ∈R−
e−δ0τ(u(·,τ)(1)
2,α +∂
∂τ u(·,τ)(−1)
0,α ).
It is complete with respect to d∗(¯u, u) := ¯u−u∗. Note that Lemmas 3.4 and 3.5
imply that ι−(a)∗≤C|a|.
Let η>0beasinCorollary3.7.Foru∈C[a],u∗≤η,letS(u;a)be a solution
of
(∂
∂τ −L)S(u;a)=E(u) (6.4)
with S(u;a)(·,0)=ι−(a)(·,0). Equivalently, we are solving
(∂
∂τ −L)(S(u;a)−ι−(a)) =E(u), <0(S(u;a)−ι−(a))(·,0)=0.
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Mean curvature flow with generic initial data 155
Existence is guaranteed by Lemma 3.2, since the a priori decay of uimplies quadratic
decay of E(u) by (3.52). Now Lemma 3.3 and (3.52)imply:
sup
τ∈R−
e−δ0τ(S(u;a)−ι−(a))(·,τ)2,α;∩BR(0)≤Cu2
∗.(6.5)
Then, (3.51), (6.5), and Lemma 3.4 imply, for τ∈R−:
e−δ0τ(S(u;a)−ι−(a))(·,τ)(1)
0
≤Ce−δ0τsup
(\BR(0))×(−∞,τ ]|E(u)|+ sup
∂BR(0)×(−∞,τ]|S(u;a)−ι−(a)|≤Cu2
∗.
(6.6)
Finally, (3.52), (6.6) and Lemma 3.5 imply, for τ∈R−:
e−δ0τ(S(u;a)−ι−(a))(·,τ)(1)
2,α
≤Ce−δ0τsup
σ≤τ(S(u;a)−ι−(a))(·,σ)(1)
0+E(u)(·,σ)(−1)
0,α ≤Cu2
∗.
Recalling also Knerr’s parabolic Schauder estimates (see Theorem B.1 in Ap-
pendix B), this implies:
S(u;a)−ι−(a)∗≤Cu2
∗.(6.7)
Therefore, S(u;a)∈C[a]. Note that solutions of (6.4) are uniquely determined
within C[a](e.g., due to Lemma 3.2). Thus, S(·,a)is a well-defined map of small
elements of C[a]into C[a].
Likewise, for ¯u∈C[a],¯u∗≤η,wehave
(∂
∂τ −L)(S(¯u;a)−S(u;a)) =E(u)−E( ¯u), <0(S(¯u;a)−S(u;a))(·,0)=0.
Therefore the discussion above applies with ¯u−uin place of u−ι−(a)and Corol-
lary 3.7’s (3.53), (3.54) instead of (3.51), (3.52), and gives:
e−δ0τ(S(¯u;a)−S(u;a))(·,τ)(1)
2,α
≤Csup
σ∈R−
e−δ0σu(·,σ)(1)
2,α +¯u(·,σ)(1)
2,α·sup
σ∈R−
e−δ0σ¯u−u(1)
2,α
i.e.,
S(¯u;a)−S(u;a)∗≤C(u∗+¯u∗)¯u−u∗.(6.8)
Consider the subset X:= {u∈C[a]:u−ι−(a)∗≤μ|a|2}. There exists μ0=
μ0(,α,δ
0)such that, for all μ≥μ0, there exists ε=ε(,α,δ0,μ) such that a∈
Bε(0)⊂RIand u∈Ximply S(u;a)∈X, by the triangle inequality and (6.7). Thus,
S(·;a)maps Xinto itself. By (6.8), it is a contraction mapping. By the completeness
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156 O. Chodosh et al.
of X, there exists a unique fixed point of S(·;a)in X, which we denote S(a).Note
that, by construction, it satisfies (6.3) and <0(S(a)(·,0)) =ι−(a)(·,0). It remains
to check that S(a)satisfies (3.45) smoothly. Indeed, E(S(a)) is Hölder continuous
in spacetime by Corollary 3.7 and Theorem B.1 in Appendix B, and the result follows
by bootstrapping standard parabolic Schauder estimates to get smoothness on S(a).
Remark Bourni–Langford–Mramor [17] recently constructed, using different meth-
ods, ancient one-sided flows coming out the Angenent torus and its higher dimen-
sional analog. Our work can be used to construct one-sided flows coming out of any
compact and any asymptotically conical shrinker.
7 Existence of a smooth ancient shrinker mean convex flow
In this section, we construct a smooth ancient shrinker mean convex flow on one
side of an asymptotically conical shrinker n⊂Rn+1. It would be possible to prove
this more in the spirit of the previous section, but thanks to the uniqueness statement
from Corollary 5.2, we can construct such a flow by any method that is convenient.
As such, we use methods that will also apply to construct a (generalized) eternal flow
which is smooth for very negative times. We will do so by modifying techniques used
in [8] to the present setting.
We fix a component of Rn+1\and assume that the unit normal to points
into . Note that by Colding–Minicozzi’s classification of entropy stable shrinkers,
[45, Theorems 0.17 and 9.36], asymptotically conical shrinkers are entropy unstable.
This (and more) is encoded in the following result:
Lemma 7.1 ([8,Propositions4.1and4.2]) The first eigenvalue μ:=λ1of the Lop-
erator (see Lemma 3.1)satisfies λ1<−1. The corresponding eigenfunction ϕ1can
be taken to be positive.For a n y β>0, it satisfies
(1+|x|2)1
2+μ−βϕ1(x)(1+|x|2)1
2+μ+β,
|∇m
ϕ1(x)|(1+|x|2)1
2+μ+β−m
2.
Moreover,there is ε0=ε0() > 0so that for ε∈(0,ε
0),the normal graph of εϕ1
is a smooth surface ε⊂.Denote ε⊂by the open set with ∂ε=ε.The
surface εis strictly shrinker mean convex to the interior of εin the sense that
2Hε+x·νε≥Cε(1+|x|2)μ
for C=C().
The following lemma is essentially [8, Proposition 4.4]. Note that because εhas
uniformly bounded curvature (along with derivatives) the time interval for which [55]
guarantees short-time existence is independent of ε→0.
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Mean curvature flow with generic initial data 157
Lemma 7.2 There is δ=δ() ∈(0,1)so that there is a smooth mean curvature flow
ε(t) for t∈[−1,−1+δ]with ε(−1)=ε.The flow remains strictly shrinker
mean convex with the bound
2tHε(t) +x·νε(t ) ≥Cε(1+|x|2+2n(t +1))μ.
We now begin the construction of an eternal weak flow that we will later prove to
have the desired properties. Fix R>0 so that for all ε∈(0,ε
0)and ρ≥1, and ε
intersect ∂BρR transversely.
Proposition 7.3 There is a smooth hypersurface ε,ρ that formed by smoothing the
corners of (ε∩BρR)∪(∂BρR ∩ε)and then perturbing slightly so that:
•as ρ→∞,ε,ρ converges smoothly on compact sets to ε,
•the level set flow of ε,ρ is non-fattening,and
•letting Kε,ρ denote the level set flow of the compact region bounded by ε,ρ ,there
is a unit-regular integral Brakke flow Mε,ρ with initial condition Mε,ρ (−1)=
Hn$ε,ρ and so that supp Mε,ρ ∩t−1((−1,∞)) =∂Kε,ρ ∩t−1((−1,∞)).
Proof Let {a
ε,ρ }a∈(−1,1)denote a foliation of smooth surfaces close to (ε∩BρR)∪
(∂BρR ∩ε)chosen so that as ρ→∞, each a
ε,ρ converges smoothly on compact
sets to ε. For all but countably many a, the level set flow of a
ε,ρ does not de-
velop a space-time interior (i.e., does not fatten); see [74, 11.3]. Write a
ε,ρ (t) :={x:
u(x,t)=a}for the corresponding level set flow. We can arrange (after re-labeling
aand changing uif necessary) that the level set flow of the pre-compact open set
bounded by a
ε,ρ is {x:u(x, t ) > a}. On the other hand, for a.e. a∈(−1,1),[74,
12.11] guarantees that7
{u=a}+=(∂∗{u>a})+,(7.1)
where Z+=Z∩t−1((−1,∞)). Assume that a=a(ε,ρ) ∈(−1,1)is chosen so that
(7.1) holds and the level set flow does not fatten.
Non-fattening guarantees that t→ Hn$∂∗{x:u(x, t ) > a }is a unit-regular inte-
gral Brakke flow Mε,ρ by [74, 11.4] (cf. [7, Theorem 3.10]). It remains to check the
condition concerning the support of both flows. Note that
(suppMε,ρ )+=⎛
⎝
t≥−1
∂∗{x:u(x,t)>a}×{t}⎞
⎠+
=(∂∗{u>a})+
={u=a}+
=(∂{u≥a})+.
7Note that in [74], there is a typo in the definition of (·)+; it is clear that the proof of [74, 12.11] only
considers points (t, x ) for tstrictly greater than the initial time.
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158 O. Chodosh et al.
The second equality is proven as in [74, 11.6(iii)], the third is (7.1) and the final
equality follows from non-fattening of a
ε,ρ . This completes the proof.
Note that we could have used the work of Evans–Spruck [59] instead of Ilmanen’s
approach [74] in the previous proof.
Lemma 7.4 There is r0=r0() > 0so that for r>r0
2,we can take ρsufficiently
large depending on rto conclude that in the space-time region
(Br\¯
Br0/2)×[−1,2],
we have that ∂Kε,ρ and Mε,ρ agree with the set flow and Brakke flow associated
to the same smooth mean curvature flow of hypersurfaces.Moreover,there is C=
C() > 0independent of rso that this flow has second fundamental form bounds
|x||A|+|x|2|∇A|+|x|3|∇2A|≤C.
Proof This follows from pseudolocality (cf. [80, Theorem 1.5]) and local curvature
estimates (cf. [53, Proposition 3.21 and 3.22]) applied on large balls far out along ε.
See also [8, Proposition 4.4].
We can now pass to a subsequential limit8ρi→∞to find a Brakke flow Mε
(resp. weak set flow Kε) with initial conditions Hn$ε(resp. Kε, the closed region
above ε; in other words, Kεis the unique closed set with Kε⊂and ∂Kε=ε).
Lemma 7.5 We have ∂Kε\t−1(−1)⊂supp Mε⊂Kε.
Proof For X∈∂Kε\t−1(−1), there is Xi∈∂Kε,ρi\t−1(−1)=supp Mε,ρi\
t−1(−1)with Xi→X. The monotonicity formula thus guarantees that X∈suppMε.
The other claim follows directly from the fact that Kεis closed.
Lemma 7.6 Take r0=r0() in Lemma 7.4 larger if necessary.Then in the space-time
region
(Rn+1\¯
Br0)×[−1,1],
both ∂Kεand Mεare the same smooth mean curvature flow which we denote by
ε(t),and satisfy
|x||A|+|x|2|∇A|+|x|3|∇2A|≤C.
Finally,ε(t ) intersects the spheres ∂Brtransversely,for all r>r
0.
Note that the smooth flows from Lemmas 7.2 and 7.6 agree when they are both
defined, so naming this flow ε(t) is not a serious abuse of notation.
8We always use Kuratowski convergence to consider limits of sets. Recall that Zn→Zif Z={x:
limsupnd(x,Zn)=0}={x:lim infnd(x,Zn)}. Because Rn+1×Ris separable, subsequential limits
in this sense always exist. See [90, §9] for further discussion.
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Mean curvature flow with generic initial data 159
Proof The smoothness and curvature estimates follow by passing the curvature es-
timates in Lemma 7.4 to the limit along a diagonal sequence r→∞. Since ∂Kε⊂
suppMε⊂Kε, we see that the smooth flows must agree. Finally, transverse intersec-
tion follows from [55, Theorem 2.1] applied to balls far out along ε=ε(−1).
Lemma 7.7 There is δ=δ() > 0so that in the space-time region
t−1([−1,−1+δ]),
both ∂Kεand Mεagree with the smooth mean curvature flow ε(t ) from Lemma 7.2.
Proof Because ε,ρiare converging smoothly to εon compact sets, pseudolocality
and interior estimates guarantee that for any r>0, there is a uniform δ>0 so that
taking isufficiently large, one component of
∂Kε,ρi∩(Br×[−1,−1+δ])
is a smooth mean curvature flow with uniformly bounded curvature (and similarly for
Mε,ρi)fort∈[−1,−1+δ].
Small spherical barriers show that for ilarge, no other component of
∂Kε,ρi∩(Br×[−1,−1+δ])
can intersect Br/2×[−1,−1+δ]. As such, sending i→∞, we can pass the curvature
estimates to the limit to find that ∂Kε∩t−1([−1,−1+δ])(and similarly for Mε)
are both smooth mean curvature flows with uniformly bounded curvature that agree
with εat t=−1. The assertion thus follows from ∂Kε⊂suppMε⊂Kεas before,
or alternatively from the uniqueness of smooth solutions to mean curvature flow with
bounded curvature, [30, Theorem 1.1].
We define the parabolic dilation map
Fλ:Rn+1×R→Rn+1×R,Fλ:(x,t)→(λx,λ
2t).
The following result is a consequence of Lemma 7.2 and relates the analytic property
of shrinker mean convexity to the behavior of the flow under parabolic dilation. It is
convenient to define
λ0:=1−δ
2
1−δ1
2
>1,(7.2)
where δis as in Lemma 7.7. Observe that Fλ(ε(t) ×{t})=λε(t ) ×{λ2t},so
λε(−λ−2)is the t=−1 slice of the parabolic rescaling (by λ) of the space-time
track of the flow t→ε(t) and the maximal smooth existence time T>−1+δ/2.
Corollary 7.8 For λ∈(1,λ
0)the surface λε(−λ−2)is contained in the interior of
ε.Moreover,for any r>0large,there is c=c(r, ) > 0so that
d(ε∩Br,λ
ε(−λ−2)∩Br)≥cε(λ −1),
for all λ∈(1,λ
0).
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160 O. Chodosh et al.
Proof By Lemma 7.2, the family of hypersurfaces defined by λ→ λε(−λ−2)has
normal speed given by
2(−λ−2)Hε(−λ−2)+x·νε(−λ−2)≥Cε(1+|x|2+2n(1−λ−2))μ.
This is strictly positive, which proves the first statement. Moreover, the speed is
strictly bounded below on Br, which proves the second statement.
For λ≥1, we define Kλ
ε:=Fλ(Kε)∩t−1([−1,1])and similarly for Mλ
ε. Recall
that λ0has been defined in (7.2). Note that ∂Kλ
ε∩t−1(−1)=λε(−λ−2)for λ∈
[1,λ
0). Below, we will write K1
ε(and similarly M1
ε) (as opposed to. Kεand Mε),
the difference being that the time parameter has been restricted to −1≤t≤1.
Lemma 7.9 There is r1=r1() > r0so that for any λ∈(1,λ
0),λε(λ−2t)\¯
Br1can
be written as the normal graph of a function ftdefined on the end of ε(t) for all
t∈[−1,1].The function ftsatisfies
|x||ft|+|x|2|∇ft|+|x|3|∇2ft|≤C,
where C=C().Moreover,
∂
∂t −ε(t) ft=a·∇ε(t)ft+bf t,
where |a|+|b|≤C=C().
Proof This follows from the argument in [8, Proposition 4.4]. Indeed, we first observe
that by taking r1sufficiently large, λε(λ−2t) and ε(t) are locally graphs of some
functions u,uλover
Bη|z|(z)⊂TzC
for η=η() > 0 and |z|>r
1sufficiently large. Differentiating the mean curvature
flow equation as in [101, Lemma 2.2] yields curvature estimates that prove that ft
exists and satisfies the asserted estimates. Finally, the fact that ftsatisfies the given
equation follows by considering the quadratic error terms when linearizing the mean
curvature flow equation; a similar argument can be found in [93, Lemma 2.5].
Proposition 7.10 The support of the Brakke flow suppM1
εis disjoint from the scaled
weak set flow Kλ
ε,for all λ∈(1,λ
0).
Proof We follow the proof of [8, Proposition 4.4], but use Ilmanen’s localized avoid-
ance principle in the compact part, due to the possible presence of singularities. Fix
λ∈(1,λ
0)and let T∈[−1,1]denote the first time the claim fails:
T=sup{τ:suppM1
ε∩Kλ
ε∩t−1((−1,τ))=∅}.
By Lemmas 7.2 and 7.7,T>−1+δ. Assume that T<1.
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Mean curvature flow with generic initial data 161
Using Theorem D.3 (recall that, by [74, 10.5] the support of a Brakke flow is a
weak set flow), we find that
suppM1
ε(T ) ∩Kλ
ε(T ) ∩B5λ0r1=∅.
Because M1
εand ∂Kεagree with the smooth flow ε(t) outside of Br0by
Lemma 7.6, there is η>0 so that9
suppM1
ε(t) ∩Kλ
ε(t) ∩(B4λ0r1\B2λ0r1)=∅,
for t∈[−1,T +η]. Now, observe that ε(t) and λε(λ−2t) are smooth flows with
the curvature estimates from Lemma 7.6 and so that the second is graphical over the
first by Lemma 7.9 (with appropriate curvature estimates). Moreover, at t=−1, the
two surfaces are disjoint, so the graphical function is initially positive.
Now, the Ecker–Huisken maximum principle [54], specifically the version in The-
orem E.1 (which applies because the graphical function satisfies the PDE given in
Lemma 7.9), to conclude that the graphical function remains non-negative for t∈
[−1,T +η](over the flow ε(t) ∩(Rn+1\B3λ0r1)). Now, the strong maximum prin-
ciple implies that the graphical function is strictly positive in ε(t) ∩(Rn+1\¯
B3λ0r1)
for t∈[−1,T +η]. Applying Theorem D.3 again, we conclude that
suppM1
ε(t) ∩Kλ
ε(t) =∅,
for t∈[−1,T +η]. This contradicts the choice of T.
Finally, we can repeat the same argument to show that the flows cannot make
contact at t=1. This completes the proof.
Corollary 7.11 For λ∈(1,λ
0),∂K1
ε\t−1(−1)is disjoint from Kλ
ε.
Proof This follows from combining Proposition 7.10 with Lemma 7.5.
Intuitively, this corollary proves that Kλ
εlies inside of K1
ε(since it has moved away
from its boundary). We make this intuition precise below. Write B◦for the interior of
asetBand Bcfor its complement.
Lemma 7.12 If A,Bare closed subsets of a topological space with Aconnected and
∂B ∩A=∅,then either A⊂B◦or A∩B=∅.
Proof We have A=(A ∩B◦)∪(A ∩Bc)∪(A ∩∂B) for any sets A,B.
Lemma 7.13 For each λ∈(1,λ
0),Kλ
ε\t−1({±1})is connected.
Proof We will prove that Kε∩t−1((−λ−2,λ
−2)) is connected for any λ∈(1,λ
0).By
Lemma 7.6, we have that
Kε∩(t−1((−λ−2,λ
−2)) \(R×Br0))
9This is just the claim that two smooth flows that are initially disjoint remain disjoint for a short time; this
holds for flows with boundary moving in any arbitrary manner.
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162 O. Chodosh et al.
is the space-time track of the region above ε(t). Hence, if Kε∩t−1((−λ−2,λ
−2))
is disconnected, then there is a connected component
R⊂Br0×(−λ−2,λ
−2).
Note that R∩t−1((−λ−2,−λ−2
0)) =∅by Lemma 7.7.
The component R“appears from nowhere,” which easily leads to a contradiction.
Indeed, we have shown that there is a point (x,t) ∈Rwith minimal t-coordinate and
because Ris a closed connected component of Kε, there is r>0 so that B2r(x)×
{t−r2}is disjoint from Kε. This contradicts the avoidance property of Kε.
Corollary 7.14 For al l λ∈(1,λ
0),Kλ
ε\t−1({±1})⊂(K1
ε)◦.
Proof This follows by combining Corollary 7.11 with Lemmas 7.12 and 7.13.
Lemma 7.15 We have (0,0)/∈Kεand for each t∈[−1,0),
suppMε(t ) ⊂√−t,
where is the open set lying above .
Proof This follows adapting of the argument [8, Proposition 4.4] to the present setting
(using Theorem D.3); as we have already given similar arguments in the proof of
Proposition 7.10, we omit the details.
We now rescale the flow as ε→0 to obtain an ancient solution. We consider
Fλ(Kε)for εsmall and λlarge (the precise relationship to be quantified in (7.3)
below) and consider this a weak set flow with initial condition λε×{−λ2}.
Lemma 7.16 For ε>0fixed,the space-time distance satisfies
lim
λ→∞d((0,0), Fλ(Kε)) =∞.
On the other hand,for λ≥1fixed,
lim
ε→0d((0,0),Fλ(Kε)) =0.
Proof The first claim follows immediately from Lemma 7.15. To prove the second
claim, it suffices (by Lemma 7.5) to show that
lim
ε→0d((0,0),supp Mε)=0.
Choose a subsequential limit ˜
Mof the flows Mεas ε→0. Note that ˜
M(−1)=
Hn$, since εconverges locally smoothly to . Using unit-regularity and unique-
ness of smooth mean curvature flows with bounded curvature (cf. [30,54]) we con-
clude via Proposition C.1 that
˜
M(t) =Hn$√−t
for t<0. This proves the claim.
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Mean curvature flow with generic initial data 163
Now, choose εi→0. It is clear that λ→ d((0,0), Fλ(Kεi)) is continuous. Thus,
for isufficiently large, we can choose λiso that
d((0,0),Fλi(Kεi)) =1.(7.3)
Taking a subsequential limit as i→∞, we find a weak set flow Kand Brakke
flow M. Note that since εi→0 we can ensure that λi→∞and thus the flow
(M,K)is ancient. We summarize the basic properties of (M,K)in the following
theorem.
Theorem 7.17 The flows Mand Khave the following properties:
(1) we have d((0,0), K)=1,
(2) theBrakkeflowMhas entropy λ(M)≤F(),
(3) we have ∂K⊂suppM⊂K,
(4) for λ>1we have Fλ(K)⊂K◦and suppM∩Fλ(K)=∅,
(5) there is T>0largesothatfort<−T,M(t) and ∂K(t ) are the same smooth
flow which we denote (t),
(6) the flow (t) lies in √−t for all t<−T,
(7) the flow (t) is strictly shrinker mean convex for all t<−T,
(8) 1
√−t(t) converges smoothly on compact sets to as t→−∞,and
(9) there is a continuous function R(t) so that,for any t∈R,
M(t)$(Rn+1\BR(t))and ∂K∩(t−1(t ) \BR(t ))
are the same smooth,multiplicity-one,strictly shrinker mean convex flow,which
we will denote by (t);moreover,there is C>0so that the curvature of
satisfies |x||A(t)|≤C.
Proof Claim (1) follows by construction. Claim (2) follows from the fact10 that
λ(ε)≤F() proven in [8, Appendix C]. The claim (3) follows as in Lemma 7.5.
We prove (4) below, but for now, we note that Corollary 7.14 immediately implies
that Fλ(K)⊂Kfor λ∈(1,λ
0). We will refer to this weaker property as (4’).
We now turn to (5). Consider M−∞, any tangent flow to Mat t=−∞. We know
that M−∞ exists and is the shrinking Brakke flow associated to an F-stationary vari-
fold V−∞ thanks to the monotonicity formula and the entropy bound λ(M)≤F().
Lemma 7.15 implies that supp V−∞ ⊂. By the Frankel property for self-shrinkers
(cf. Theorem D.4), it must hold that ∩supp V−∞ = ∅. By the strong maximum
principle for stationary varifolds [77,97] (either result applies here because is
smooth), there must exist a component of suppV−∞ which is equal to . By the con-
stancy theorem (and Frankel property again) we find that V−∞ =kHn$,forsome
integer k≥1. By the entropy bound in (2), k=1. Thus, by Brakke’s theorem (c.f.
[108]) and Lemma 7.18, there is T>0 large so that M(t) is the multiplicity one
Brakke flow associated to a smooth flow (t) (and 1
√−t(t) converges smoothly on
compact sets to as t→−∞). Since ∂K⊂supp M, we see that ∂K(t) =(t) as
well. This completes the proof of (5); note that we have proven (8) as well.
10Note that the simpler statement λ(ε)≤F()+o(1)as ε→0 would suffice here.
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164 O. Chodosh et al.
By Lemma 7.15,(t) ⊂√−t¯
. Since √−t and (t) are both smooth (for
t<−T), they cannot touch unless (t) =√−tfor all t<−T. This cannot happen
by an argument along the lines of Lemma 7.16. This proves (6).
Now, we note that (4’) implies that (t) is weakly shrinker mean convex. By the
strong maximum principle (see [96, Proposition 4] for the evolution equation for the
shrinker mean curvature), (t) is either a shrinker for all t<−Tor strictly shrinker
mean convex. The first case cannot occur (by the argument used for (6)), proving (7).
By Lemma 7.18 proven below, we know that for tsufficiently negative, 1
√−t(t)
is an entire graph over of a function with small ·(1)
3norm. From this, we can
use pseudolocality to prove (9) exactly as in [8, Proposition 4.4(1)] (the exterior flow
M(t)$(Rn+1\BR(t))=(t) is weakly shrinker mean convex by (4’) and thus strictly
so by the strong maximum principle).
Finally, we prove (4). Strict shrinker mean convexity of the exterior flow guaran-
tees that for λ>1, supp Mand Fλ(K)are disjoint outside of a set Din space-time
which has D∩t−1([a,b])compact for any a<b. Thus, we may apply Ilmanen’s
localized avoidance principle, Theorem D.3, to show that suppMand Fλ(K)are
indeed disjoint. Using (3) and (4’), this completes the proof of (4).
The following lemma was used above, and we will also use it again when proving
uniqueness of ancient one-sided flows.
Lemma 7.18 Suppose that (S(τ ))τ≤0is an ancient rescaled mean curvature flow so
that S(τ) converges to smoothly with multiplicity one on compact sets as τ→−∞.
Then,for τsufficiently negative,there is a function u(·,τ) on so that S(τ) is the
normal graph of u(·,τ) over and so that
lim
τ→−∞u(·,τ)(1)
3=0.
Proof This follows from an simplified version of the argument used in [31,
Lemma 9.1]. Indeed,
ˆ
Sτ0(t) :=√−tS(τ
0−log(−t))
is an ancient mean curvature flow for t≤−eτ0. Moreover, as τ0→−∞the flows
(ˆ
Sτ0(t))t≤−eτ0converge smoothly on any compact subset of ((−∞,0]×Rn+1)\
{(0,0)}to the shrinking flow {√−t}t≤0(cf. the proof of Lemma 7.15). This implies
that there is τ1sufficiently negative so that for τ0<τ
1and t∈[−1,−eτ0],
ˆ
Sτ0(t) ∩(B2r\Br)
is the graph of some smooth function ˆuτ0defined on a subset of √−t and that
sup
t∈[−1,−eτ0]ˆuτ0(·,t)C3→0 (7.4)
as τ0→−∞. Below, we will always assume that τ0<τ
1.
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Mean curvature flow with generic initial data 165
We can rescale the above observation back to S(τ ) to find that
S(τ) ∩x:re
τ−τ0
2≤|x|<2re
τ−τ0
2
is the graph of some function u(·,τ) defined on some r,τ0(τ ) ⊂, as long as τ0∈
(−∞,τ]and τ≤τ1. For such τ, by varying τ0∈(−∞,τ], we find that S(τ ) \Bris
the graph of some function u(·,τ) over the domain
r(τ ) :=
τ0∈(−∞,τ ]
r,τ0(τ ) ⊂.
Shrinking τ1if necessary, we can assume that
\B2r⊂r(τ ),
for τ<τ
1by the smooth convergence of S(τ) to on compact sets.
Finally, the C3estimate (7.4) rescales as follows. We have
u(·,τ)(1)
3;r,τ0(τ ) ˆuτ0(·,−eτ0−τ)C3
for τ0∈(−∞,τ]. In particular, we find
u(·,τ)(1)
3;r(τ ) =sup
τ0∈(−∞,τ ]u(·,τ)(1)
3;r,τ0(τ )
sup
τ0∈(−∞,τ ]ˆuτ0(·,−eτ0−τ)C3
≤sup
τ0∈(−∞,τ ]
sup
t∈[−1,−eτ0]ˆuτ0(·,t)C3.
For τsufficiently negative, u(·,τ) extends across the compact part of with C3-
norm tending to 0, so combined with the previous inequality and (7.4), the result
follows.
8 Long-time regularity of the flow
In this section, we analyze further the flow (M,K)from Theorem 7.17.Wemust
separate our analysis into three time scales, t<0, t=0, t>0.
8.1 Regularity for t<0
Here, we show that White’s regularity theory [105,107] for mean-convex flows ap-
plies to the flow (M,K)for t<0.
Remark Because it plays a fundamental role in our analysis, we briefly recall White’s
strategy. The basic setup is to prove that the Brakke flow and level-set flow are com-
patible in the sense that the Brakke flow is the n-dimensional Hausdorff measure
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166 O. Chodosh et al.
restricted to the boundary of the level-set flow. This lets White combine Brakke flow
(density/monotonicity) arguments with the fact that the level set flow is moving to one
side. In particular, the one-sidedness implies that the level set flow is minimizing to
the outside. The key to White’s regularity is then to rule out multiplicity two planes as
static/quasi-static tangent flows (higher multiplicity cannot occur by the minimizing
property and other tangent flows are less common and thus less troublesome thanks
to stratification). Using the fact that the flow is moving to one-side and a “no holes”
argument, White then proves that such a tangent flow will locally separate into two
sheets. Then, linear analysis can be used to rule out such a situation.
We define the rescaled flow ˜
K(and analogously for ˜
M)by
˜
K:=
τ∈(−∞,∞)
eτ
2K(−e−τ)
for τ∈(−∞,∞). It is easy to see that ˜
Kis still a closed subset of space-time. Indeed,
it is the image of a closed set under the diffeomorphism
R:Rn+1×(−∞,0)→Rn+1×R,R:(x,t)→((−t)−1
2x,−log(−t)).
Remark The rescaled flows will be seen to be moving to one side (like a mean convex
mean curvature flow). Because the rescaling is “lower order” and in particular disap-
pears after taking a blow-up, White’s theory will apply to the rescaled flows as well.
Here, a serious issue will be that we do not aprioriknow compatibility of the level
set flow and the Brakke flow (we only know “partial’ compatibility, cf. (3) in Theo-
rem 7.17). As such, we will combine White’s theory with a continuity argument to
work up until the first time the theory breaks down (cf. (8.2)). A crucial observation
is that White can rule out static/quasi-static multiplicity tangent flows at some time
¯τusing knowledge of the flow only for prior times τ<¯τ(of course, this is simply a
manifestation of the parabolic nature of the flow).
Let
Th:Rn+1×R→Rn+1×R,Th:(x,t)→(x,t −h) (8.1)
denote the time-translation map.
Lemma 8.1 For h>0, we have
Th(˜
K)⊂˜
K◦and supp ˜
M∩Th(˜
K)=∅.
Proof Note that Th(˜
K)=R(Feh
2(K)). Thus, (4) in Theorem 7.17 implies both
claims.
Proposition 8.2 We hav e ∂˜
K=supp ˜
M.
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Mean curvature flow with generic initial data 167
Proof Suppose that (x,τ)∈supp ˜
M\∂˜
K⊂˜
K◦. Choose r>0 so that Br(x)⊂˜
K(τ ).
Lemma 8.1 implies that Br(x) is disjoint from supp ˜
M(τ −h) for all h>0 small.
For hsufficiently small, the rescaled level set flow Bgenerated by Br(x)×{τ−h}
has (x,τ)∈B◦. On the other hand, supp M∩B=∅by the avoidance principle. In
particular, (x,τ) /∈suppM. This is a contradiction.
Proposition 8.2 and [74, 10.5] imply that ∂˜
Kis a (rescaled) weak set flow.
Corollary 8.3 If (x,τ
0)∈reg ˜
Mthen there is r>0so that
˜
M(τ )$Br(x)=Hn$(∂ ˜
K(τ ) ∩Br(x))
for τ∈(τ0−r2,τ
0+r2),and ˜
K◦∩(Br(x)×(τ0−r2,τ
0+r2)) =∅.
Proof By definition, there is r>0 sufficiently small so that ˜
M(τ )$Br(x)=
Hn$˜
M(τ) for ˜
M(τ) a smooth rescaled mean curvature flow in Br(x). Thus,
∂˜
K∩(Br(x)×(τ0−r2,τ
0+r2)) =supp ˜
M∩(Br(x)×(τ0−r2,τ
0+r2))
=
|τ−τ0|<r2˜
M(τ) ×{τ}.
This proves the first statement. The second statement follows from Lemma 8.1 and
Proposition 8.2.
Corollary 8.4 For τ0∈R,we have (∂ ˜
K)∩{τ=τ0}=∂( ˜
K∩{τ=τ0}).
As such, we can (and will) unambiguously write ∂˜
K(τ0)for either of these sets.
Proof It is clear that
(∂ ˜
K)∩{τ=τ0}⊃∂( ˜
K∩{τ=τ0}),
and that
(∂ ˜
K)∩{τ=τ0}⊂(˜
K∩{τ=τ0}).
Consider now
x∈(∂ ˜
K)∩{τ=τ0}∩(˜
K∩{τ=τ0})◦.
Considering a small shrinking ball from a slightly earlier time, as in the proof of
Proposition 8.2, we see that (x , τ ) ∈˜
K◦, a contradiction.
Lemma 8.5 The sets {∂˜
K(τ )}τ∈Rform a singular foliation of .
Proof Note that the sets {∂˜
K(τ )}τ∈Rare disjoint by Lemma 8.1. Now, note that
lim
h→−∞Th(˜
K)=, lim
h→∞Th(˜
K)=∅,
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168 O. Chodosh et al.
by Theorem 7.17. As such, for x∈, we can choose the maximal T∈Rso that
(x, T ) ∈˜
K. Assume that (x, T ) ∈˜
K◦. By considering a small shrinking ball barrier
as in the proof of Proposition 8.2, we can contradict the choice of T.
Recall that the F-area of a measure μ(with μ(Br)rkfor some k>0) is
F(μ) := (4π)−n
2e−1
4|x|2dμ(x)
(cf. Sect. 2.7.) Set also F(A):= F(Hn$A) when it is defined. We have the following
proposition, which is a straightforward modification of the corresponding result in
the mean-convex case.
Proposition 8.6 (cf. [105,Theorems3.5,3.8,and3.9]) Suppose that Vis a locally
F-area minimizing hypersurface (integral current)contained in with boundary in
˜
K(τ ).Then V⊂˜
K(τ ).In particular,∂˜
K(τ ) has locally finite Hn-measure and for
any Br(x)⊂,
F(∂ ˜
K(τ ) ∩Br(x)) ≤F(∂B
r(x)).
Finally,for Br(x)⊂,if Sis a slab of thickness 2εr passing through xand
∂˜
K(τ ) ∩Br(x)⊂Sthen ˜
K(τ ) ∩(Br(x)\S) consists of k=0,1, or 2of the con-
nected components of Br(x) \Sand
F(∂K(τ ) ∩Br(x)) ≤(2−k+2nε +e(r ))ωnrn,
where e(r) =o(1)as r→0.11
At this point, we have no guarantee that the Brakke flow ˜
Mhas ˜
M(τ ) =
Hn$∂˜
K(τ ) as in [105, §5]. As such, we cannot immediately deduce regularity fol-
lowing [105,107]. Instead, we must use a continuity argument: consider the set in
space-time
D:={X∈Rn+1×R:˜
M(X) ≥2}.(8.2)
By upper semi-continuity of density, it is clear that Dis closed. Moreover, by (5) in
Theorem 7.17, it is clear that the projection of Donto the τ-axis is bounded from
below, and the projection on Rn+1-factor is bounded. As such, if Dis non-empty, we
can choose an element ¯
X=(¯
x,¯τ) ∈Dwith smallest possible τ-coordinate.
Lemma 8.7 (cf. [105, Theorem 5.5]) If (˜
Mi,˜
Ki)is a blow-up sequence limiting to
(˜
M,˜
K),around points (xi,τ
i)with lim supi→∞ τi<¯τthen supp ˜
M=∂˜
Kand
∂˜
K
i→∂˜
K.
Proof As usual, we can show that ∂˜
K⊂supp ˜
M⊂˜
K. On the other hand, by [104,
§9], almost every X∈supp ˜
Mhas a tangent flow that is a static or quasi-static plane.
11We emphasize that this last statement does not hold uniformly for all x.
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Mean curvature flow with generic initial data 169
By definition of ¯τ, these must be static and multiplicity-one (by unit regularity). Thus,
Corollary 8.3 implies that there must be points in the complement of ˜
Kthat are
arbitrarily close to X, since (˜
Mi,˜
Ki)converges smoothly near X. This implies that
a dense subset of supp ˜
Mis contained in ∂˜
K. This completes the proof.
Lemma 8.8 ([105, Theorem 7.2]) If (˜
M,˜
K)is a static or quasi-static limit flow at
(x, τ ) with τ<¯τ,then ˜
Mis a stable minimal hypersurface whose singular set has
Hausdorff dimension at most n−7. In particular,a non-flat static or quasi-static limit
flow cannot exist when n<7.
From now on, we assume that n<7.
Corollary 8.9 We have (sing ˜
M)∩{τ<¯τ}is of parabolic Hausdorff dimension ≤
n−1. Moreover,for each τ<¯τ,at time τ,the singular set sing ˜
M(τ ) has spatial
Hausdorff dimension at most n−1.
Proof This follows from Lemma 8.8 and [104, §9]. See also [105, Theorem 1.3].
Corollary 8.10 For τ<¯τ,˜
M(τ ) =Hn$∂˜
K(τ ).
Proof Corollary 8.9 implies that Hn(supp ˜
M(τ ) \reg ˜
M(τ )) =0. Because ˜
Mhas
bounded entropy, we have that ˜
M(τ )(Br(x)) rnwhich implies that
˜
M(τ )(supp ˜
M(τ ) \reg ˜
M(τ )) =0.
Combined with supp ˜
M(τ ) =∂˜
K(τ ), the assertion follows.
Proposition 8.11 The set Dis empty.Moreover,for any limit flow (M,K),we have
that suppM=∂Kand there is T≤∞so that
(1) K(t) is weakly convex for all t,
(2) K(t) has interior points if and only if t<T,
(3) ∂K(t) are smooth for t<T,
(4) M(t) is smooth and multiplicity one for t<T,
(5) K(t) is empty for t>T.
If (M,K)is a tangent flow,then it is a multiplicity one generalized cylinder Sn−k×
Rk.
Proof We first prove that Dis empty by arguing that we can apply the regularity the-
ory of [105,107]at ¯τ. Observe that Lemma 8.1, Proposition 8.6, and Corollary 8.10
allow us to apply all of the arguments in [105] that do not consider any points from
{τ≥¯τ}(after this time we do not know how to relate ˜
Mand ˜
K).
Assuming D=∅, we can fix (¯
x,¯τ) ∈D.Let(M,K)denote a tangent flow pair
to (˜
M,˜
K). Arguing as in [105, Theorem 5.5] we find that Mis compatible with the
associated weak set flow Kfor times t<0 and the rescalings of ∂˜
Karound (¯
x,¯τ)
converges to ∂Kon {t<0}as sets. In particular, this allows us to apply the arguments
in [105, §9] to conclude that (M,K)cannot be a multiplicity-two hyperplane for
t<0 (either static or quasistatic).
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170 O. Chodosh et al.
Remark Note that [105, §9] considers multiplicity-two quasistatic planes (and static
planes were already ruled out in [105, Corollary 8.5]). We cannot appeal to [105,
Corollary 8.5] in this setting, since the argument would need information about the
flow for times τ>¯τ. However, one may carefully check that the argument in [105,
§9] makes no reference to any time τ>¯τnor does it need [105, Corollary 8.5], just
the sheeting theorem [105, Theorem 8.2] which is applied to the blow-up sequence on
compact subsets of Rn+1×{t<0}. (In particular, we are making the observation that
the argument in [105, §9] can be used to rule out static (or quasi-static) multiplicity-
two planes while only considering times before the singular time.)
Now that we have seen that (M,K)cannot be a multiplicity-two hyperplane for
t<0, we claim that sing M∩{t<0}=∅. If not, there is some X∈sing M∩{t<
0}. An iterated tangent flow (M,K)at Xwill be static (since t<0) and is a
limit flow of (˜
M,˜
K)(and will only see points at <¯τ), cf. [105, Theorem 5.2(1)].
Thus, we can repeat the argument in [105, Theorem 12.3] to show that (M,K)
cannot be a multiplicity-two plane. (Because (M,K)is not a multiplicity-two plane,
the rescaling chosen in the proof [105, Theorem 12.3] will still yield a tangent flow
to (M,K)and will thus not see any points with τ>¯τ.) Iterating this argument
and ruling out a non-trivial union of half-planes using Proposition 8.6 as in [105,
Theorem 7.2], we can conclude (using the assumption n<7) that (M,K)is a
multiplicity-one hyperplane, contradicting X∈sing M. Since (M,K)is regular
for t<0, it must be a generalized cylinder by (mean) convexity, [107, Theorem 10]
(cf. [42, Theorem 10.1]). In particular, this implies that ˜
M(¯
x,¯τ) < 2, contradicting
the definition of D.
Now that D=∅, we can use Lemma 8.1, Proposition 8.6, and Corollary 8.10 to
see that White’s regularity theory [105,107] applies to (˜
M,˜
K)for all time. This
completes the proof.
We will say that (x,t) ∈singMhas a mean convex neighborhood if12 there’s
ε>0 so that t+ε2<0 and if t−ε2<t
1<t
2<t+ε2then
K(t2)∩Bε(x)⊂K(t1)∩Bε(x)\∂K(t1).
With this definition, we can now summarize the above conclusions for the non-
rescaled flow.
Corollary 8.12 The non-rescaled flows (M,K)have the following properties for
t<0
(1) M(t) =Hn$∂K(t),
(2) singM∩{t<0}has parabolic Hausdorff dimension ≤n−1and for t<0,
singM(t ) has spatial Hausdorff dimension ≤n−1,
(3) any limit flow at X=(x,t) with t<0is weakly convex on the regular part and
all tangent flows are multiplicity one generalized cylinders,and
(4) any singular point has a (strict)mean-convex neighborhood.
12This definition is slightly simpler than the one used in [39,89] since all singularities in our setting have
the “same orientation” (since the shrinker mean convexity rescales to mean convexity in the blow-up limit).
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Mean curvature flow with generic initial data 171
Proof Everything but the last claim is proven above (in the rescaled setting). The last
claim follows from the fact that all limit flows are convex so [89] applies.
8.2 Regularity at t=0
We now turn to regularity near time t=0.
For A, B ⊂Rn+1×R, subsets of space-time, we write
dE(A, B ) =inf
(xa,ta)∈A,(xb,tb)∈B|xa−xb|2+(ta−tb)2
for the Euclidean distance between the two sets. We emphasize that this differs from
the usual parabolic distance between the sets. Note that the parabolic dilation map
Fλ:Rn+1×R→Rn+1×Rgenerates the vector field
V:= d
dλλ=1Fλ=(x,2t) ∈TxRn+1⊕TtR.
We now consider the geometry of hypersurfaces in space-time swept out by a mean-
curvature flow.
Lemma 8.13 Consider a family of smooth hypersurfaces (a, b) → M(t) ⊂Rn+1
flowing by mean curvature flow.Set
M:=
t∈(a,b)
M(t) ×{t}.
Then,Mis a smooth hypersurface in spacetime Rn+1×Rwith unit normal13 at (x,t)
given by
νM=νM(t) +HM(t)(x)∂t
1+HM(t)(x)2.
Moreover,the normal speed of λ→Fλ(M)at λ=1is
2tHM(t) +x·νM(t)
1+HM(t)(x)2.(8.3)
Proof The given unit vector is orthogonal to
T(x,t ) M=TxM(t)⊕spanR(∂t+HM(t)(x)).
This implies the expression for νM. To prove (8.3), we may compute
V·νM=(x+2t∂t)·(νM(t) +HM(t)(x)∂t)
1+HM(t)(x)2=2tHM(t) +x·νM(t)
1+HM(t)(x)2.
This completes the proof.
13We emphasize that the unit normal is taken with respect to the Euclidean inner product on spacetime
Rn+1×RRn+2.
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172 O. Chodosh et al.
Now, recall that by Theorem 7.17, there is a smooth flow (t) so that ∂K(t)
and M(t) agree with (t) outside of BR(t) and on Rn+1×(−∞,−T). Choose R0
sufficiently large so that R0≥R(t) for t∈[−4T,0](we will take R0larger in (8.4)
in Proposition 8.15 below). Then, define
S:=⎛
⎝
−4T≤t≤−2T
((t ) ∩¯
B3R0)×{t}⎞
⎠∪⎛
⎝
−2T≤t≤1
(t) ∩(¯
B3R0\BR0)×{t}⎞
⎠.
Lemma 8.14 There is c=c(R0,),C(R0,) > 0and λ1=λ1(R0,)>1so that
c(λ −1)≤dE(S,Fλ(S)) ≤C(λ −1)
for λ∈(1,λ
1).
Proof It suffices to show that
d
dλλ=1dE(S,Fλ(S)) ∈(0,∞).
This follows from (8.3) (and the compactness of S) since positivity of the shrinker
mean curvature of (t) was established as (7) and (9) in Theorem 7.17.
Proposition 8.15 For r>0sufficiently large,there is c=c(r, ) > 0and λ
1=
λ
1(r, ) so that
dE(∂K∩(¯
Br×[−1,0]), Fλ(∂ K)∩(¯
Br×[−1,0])) ≥c(λ −1)
for λ∈(1,λ
1).
Proof Given r>0 large, we fix R0by requiring that
4R2
0+6nT ≥4r2(8.4)
and that R0≥R(t) for t∈[−3T,0](where R(t) is defined in Theorem 7.17). This
choice of R0will allow us to use Theorem D.3 below. We fix c=c(R0,) as in
Lemma 8.14 and will choose c&cbelow.
For λ−1>0 sufficiently small, assume that
dE(∂K∩(¯
Br×[−1,0]), Fλ(∂ K)∩(¯
Br×[−1,0])) < c
2(λ −1)(8.5)
(otherwise the assertion follows) and that the distance is achieved at
(x,t)∈∂K∩(¯
Br×[−1,0]), (x+z,t +s) ∈Fλ(∂ K)∩(¯
Br×[−1,0]).
In particular, |s|≤c
2(λ −1).
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Mean curvature flow with generic initial data 173
Recalling the translation map Tsdefined in (8.1), observe that, Lemma 8.14 and
(8.5) imply that
dE(Ts(Fλ(S)), S)≥dE(Fλ(S), S)−dE(Ts(Fλ(S)), Fλ(S))
≥dE(Fλ(S), S)−|s|
≥c
2(λ −1).
Consider the weak set flows Ts(Fλ(∂K)) and ∂K. From the previous estimate and
Theorem D.3 with a=t0=−3T,b=t,R=2R0,x0=0, and γsmall we see that
Ts(Fλ(∂K)) and ∂Kare disjoint for t∈[−3T,0]. Recall that in Theorem D.3 the dis-
tance dt(see (D.2)) is defined with the choice u=(R2−|x−x0|2−2n(t −t0))+.(8.4)
implies that uis uniformly bounded from below away from zero on ¯
Br×[−3T,0],
so Theorem D.3 allows further to conclude that (here and below, the implied constant
in ,depend on r,R0,but not on λand t)
|z|dt(Ts(Fλ(∂K)), ∂ K)≥d−3T(Ts(Fλ(∂K)), ∂ K).
However, by Lemma 8.14, and since |u|≤4R2
0, we see that
d−3T(Ts(Fλ(∂K)), ∂ K)c(λ −1)−|s|.
Putting these inequalities together, we find that
c(λ −1)|z|+|s|=dE(∂ K∩(¯
Br×[−1,0]), Fλ(∂ K)∩(¯
Br×[−1,0])).
This completes the proof.
Corollary 8.16 For r>0, there is s=s(r, ) > 0with the following property.Choose
(x,t)∈reg M∩Br×[−1,0]and fix a space-time neighborhood Uof (x,t) so that
in U,Magrees with t→Hn$M(t),for a smooth mean curvature flow M(t).Then,
2tHM(t)(x)+x·νM(t)(x)≥s1+HM(t)(x)2.
Proof Proposition 8.15 implies that the speed of λ→ λM (λ−2t) at λ=1 has a uni-
formly positive lower bound. Thus, the conclusion follows from (8.3).
Corollary 8.17 There is C=C() > 0and δ=δ() ∈(−1,0)so that ∂K(t) is
smooth with |H∂K(t)|≤Cfor t∈(δ , 0).
Proof By (9) in Theorem 7.17 it suffices to prove this for points in Brfor some r>0
sufficiently large. Fixing such an r, Corollary 8.16 implies that there is s>0 so that
2tHM(t)(x)+x·νM(t)(x)≥s1+HM(t)(x)2
for (x,t)∈reg M∩Br×[−1,0]. Solving for H, we find that |H|≤Con regM∩
(Br×(−2δ,0)for some δ∈(−1,0)sufficiently small.
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174 O. Chodosh et al.
However, by (3) in Corollary 8.12,anyX∈sing M∩{t<0}has a multiplicity-
one generalized cylinder as a tangent flow. In particular, there are points Xi∈reg M∩
{t<0}with Xi→Xand H(X
i)→∞. This contradicts the mean curvature bound,
completing the proof.
Corollary 8.18 We have that singM(0)=∅,M(0)=Hn$∂K(0)and x·ν∂K(0)>0,
Proof By Corollary 8.17, we know that for t∈(δ, 0)and x∈∂K(t),|H∂K(t ) (x)|≤C.
Thus, by Corollary 8.16, we conclude that for rchosen as in the proof of Corol-
lary 8.17, taking δsmaller if necessary, for t∈(δ, 0)we find that ∂K(t) is strictly
star-shaped in Br, i.e., there is c>0 so that
x·ν∂K(t) ≥c
for x∈∂K(t) ∩Br. In particular, this implies that ∂K(t) is locally uniformly graph-
ical. Interior estimates [55, Theorem 3.1] then imply that the flow ∂K(t) remains
smooth and strictly star-shaped up to t=0 (outside of Br, the flow is automatically
smooth and strictly star-shaped by (7) and (9) in Theorem 7.17).
8.3 Regularity for t>0
Using sing M(0)=∅and (9) from Theorem 7.17, there is some ˆ
δ>0 so that M(t) =
Hn$∂K(t) is smooth for t∈[0,ˆ
δ). We can now consider the rescaled flow
ˆ
K:=
τ∈(−∞,∞)
e−τ
2K(eτ),
and ˆ
Msimilarly defined, exactly as in the t<0 situation. The only difference is that
the flow is moving outwards rather than inwards:
Th(ˆ
K)⊂ˆ
K◦
for h<0 (cf. Lemma 8.1). This does not seriously affect the arguments used above,
and we find that Corollary 8.12 holds for t>0 as well.
8.4 Long time asymptotics
We continue to use our notation from the t>0 regularity section. Moreover, we de-
note with Cthe asymptotic cone of the asymptotically conical shrinker. We will
also need to consider the integral unit-regular Brakke flows t∈[0,∞)→μ±(t ) con-
structed in Theorem F.2 whose support agrees with the inner and outer flow M±(t)
of C. They can be used to prove:
Lemma 8.19 For al l t≥0, ∂K(t) is disjoint from the level set flow of C.
Proof Note that (μ±(t))t≥0is smooth with unit multiplicity outside of B√tR0(0)for
some R0>0. Moreover, μ±(0)is disjoint from Cat t=0. Thus, we can argue as
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Mean curvature flow with generic initial data 175
in Proposition 7.10: we may couple the Ecker–Huisken Maximum Principle (Theo-
rem E.1), with Ilmanen’s localized avoidance principle (Theorem D.3) to show that
∂K(t) is disjoint from M±(t ) for all t≥0. This implies the claim.
This allows to characterize the convergence of the rescaled flow for τ→∞.We
assume that M(t) lies outside the outer flow M+(t) of the level set flow of C.
Theorem 8.20 The rescaled flow ˆ
M(τ ) converges smoothly as τ→∞to an ex-
pander E,which is smoothly asymptotic to Cand minimizes the expander func-
tional
E(S) =S
e1
4|x|2dHn(8.6)
from the outside (relative to compact perturbations)and is thus smooth.Furthermore,
M+(t) =√tE
for t>0.
Proof Since τ∈(0,∞)→ ˆ
M(τ ) is expander mean convex, and is smooth with uni-
form control on all derivatives outside of BR0(0), it follows from the arguments in
[105, §11], that ˆ
M(τ ) converges smoothly to an outward minimizing minimal sur-
face Ein the expander metric g=e1
2n|x|2gRn+1. This yields the claimed regular-
ity and the smoothness of the convergence. Note that any blow down of the flow
t∈[0,∞)→ M(t) lies inside the level set flow of C,soEhas to be smoothly
asymptotic to C. By Lemma 8.19 the flow t→ √tE has to agree with the outer
flow of C.
8.5 The outermost flows of general hypercones
We consider, for n<7, a general embedded, smooth hypersurface ⊂Snand the
regular hypercone C() ⊂Rn+1. We show in this subsection that the previous ar-
guments can be generalized to characterize the outer and inner flows of the level set
flow of C() as in Theorem 8.20.
Note that divides Sninto two open sets S±. We can construct smooth hyper-
surfaces M±which are smooth radial graphs over S±, smoothly asymptotic to C()
with sufficiently fast decay such that x·νM±(with νM±the upwards unit normal)
decays to zero at infinity along M±.Let(M±(t))t∈[0,T ±)be the maximal smooth
evolution of M±. Note that by the maximum principle of Ecker–Huisken [54]to-
gether with the strong maximum principle we have that
2tHM±(t) +x·νM±(t ) >0
along (M±(t))t∈(0,T ±). We can thus repeat the arguments in Sect. 8.3 to construct
expander mean convex flows (M±(t))t>0such that the corresponding rescaled flows
converge to expanders, smoothly asymptotic to C(). This implies
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176 O. Chodosh et al.
Theorem 8.21 The outermost flows of C() are given by expanding solutions t∈
(0,∞)→ √tE±smoothly asymptotic to C.The expanders E±minimize the ex-
pander energy (8.6)from the outside (relative to compact perturbations)and are
smooth.
See also the notes of Ilmanen [75] for the proof of smoothness in case n=2.
Furthermore by an argument of Ilmanen–White [75] any such outermost expander
has genus zero.
9 Uniqueness and regularity of one-sided ancient Brakke flows
We now combine the three regimes considered above with Theorem 7.17 to conclude
the following existence and regularity for the flow (M,K).
Theorem 9.1 (One-sided existence) Fo r n≤6and na smooth asymptotically con-
ical self-shrinker,choose a fixed component of Rn+1\.Then,there exists an
ancient unit-regular integral Brakke flow Mand weak set flow Kwith the following
properties:
(1) M(t) =Hn$∂K(t),
(2) ∂K(t) ⊂√−t for all t<0,
(3) there is T>0so that for t<−T,M(t) is a smooth multiplicity one flow (t)
with (t) is strictly shrinker mean convex,
(4) 1
√−t(t) converges smoothly on compact sets to as t→−∞,
(5) there is a continuous function R(t) so that for any t∈R,M(t)$(Rn+1\BR(t))
is a smooth strictly shrinker mean convex multiplicity one flow (t),
(6) theBrakkeflowMhas entropy λ(M)≤F(),
(7) singMhas parabolic Hausdorff dimension ≤n−1and for any t∈R,
singM(t ) has spatial Hausdorff dimension ≤n−1
(8) any limit flow is weakly convex on the regular part and all tangent flows are
multiplicity one generalized cylinders,
(9) any singular point has a strictly mean-convex neighborhood,
(10) there is δ>0so that ∂K(t) is completely smooth for t∈(−δ, δ) and ∂K(0)is
strictly star-shaped,and
(11) 1
√t∂K(t) converges smoothly on compact sets to an outermost expander coming
out of the cone at infinity of ,as t→∞.
Now, we will combine Theorem 9.1 with Corollary 5.2 to prove uniqueness of the
flow constructed above.
Theorem 9.2 (One-sided uniqueness) Fo r n≤6, fix na smooth asymptotically
conical self-shrinker as in Theorem 9.1.Let (μt)−∞<t<∞be a unit-regular integral
Brakke flow such that
suppμtis strictly on one side of √−t for every t∈(−∞,0), (9.1)
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Mean curvature flow with generic initial data 177
and
0< ((μt), −∞)<2λ(). (9.2)
After a time translation,μtcoincides with the Brakke flow from Theorem 9.1.
Proof As in the proof of (5) in Theorem 7.17, the Gaussian density bound guarantees
that the tangent flow to μtat −∞ is the multiplicity one shrinker associated to .
As such, Lemma 7.18 and Corollary 5.2 imply that after a time-translation there is
T>0 so that for t≤−T,μt=Hn$(t), where (t) is the smooth flow from The-
orem 9.1 (4).
As in Proposition 7.10, Ilmanen’s localized avoidance principle (Theorem D.3)
combined with Ecker–Huisken’s maximum principle at infinity (Theorem E.1), we
see that supp μtis disjoint from Fλ(suppM)for λ= 1. This implies that suppμt⊂
suppM.
Finally, since reg Mis connected by (9) in Theorem 9.1 and Corollary G.5,wesee
that μt=M(t) in (sing M)c(using the unit-regularity of Mand μt). This completes
the proof.
Remark Both Theorems 9.1 and 9.2 clearly hold (with simpler proofs) in the case
that is a smooth compact shrinker.
Remark We expect that the dimensional restriction in Theorems 9.1 and 9.2 can be
removed (cf. [56,62,111]). We note that when has sufficiently small F-area,
Theorems 9.1 and 9.2 hold in all dimensions. See §10 for a precise statement.
10 Generic mean curvature flow of low entropy hypersurfaces
We recall the following notions from [11]. Denote by Snthe set of smooth self-
shrinkers in Rn+1and S∗
nthe non-flat elements. Let
Sn() :={∈Sn:λ() < }
and similarly for S∗
n().LetRMCndenote the set of regular minimal cones in Rn+1
and define RMC∗
n,RMCn(),RMC∗
n() analogously. We now recall the follow-
ing two “low-entropy” conditions from [11]:
RMC∗
k() =∅ for all 3 ≤k≤n(n, )
and
S∗
n−1() =∅.(n,)
It’s convenient to set λk=λ(Rn−k×Sk).
Given these definitions, we can state the following result.
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178 O. Chodosh et al.
Theorem 10.1 For n≥3and ∈(λn,λ
n−1],assume that (n,)and (n, )hold.
Then if M⊂Rn+1is a closed hypersurface with λ(M) ≤there exist arbitrarily
small C∞graphs Mover Mand corresponding unit-regular integral Brakke flows
Mwith M(0)=Hn$M,so that Mis completely regular until it disappears in a
round point.That is,there is X∈Rn+1×Rso that sing M={X}and so that any
tangent flow at Xis a round shrinking Sn.
We will prove this below. Note that (3,λ2)holds by [86, Theorem B] and (3,λ2)
holds by [8, Corollary 1.2], so Theorem 10.1 implies Theorem 1.1.
We also note that Theorems 9.1 and 9.2 hold in all dimensions with the assump-
tion that (n,)holds and F()≤. Indeed, the dimension restriction in Theorems
9.1 and 9.2 arises due to the use of [107], where it is used to rule out static cones
as limit flows to a mean-convex flow (cf. [107, Theorem 4]). However, in the low-
entropy setting static cones cannot occur as limit flows, by assumption (n,)(cf.
[11, Lemma 3.1]) even without assuming mean-convexity.
Lemma 10.2 ([11, Proposition 3.3]) Fo r n≥3and ∈(λn,λ
n−1],assume that
(n,)and (n, )hold.If Mis a unit-regular integral Brakke flow with λ(M)≤
then any tangent flow to Mis the multiplicity one shrinker associated to a smooth
shrinker that is either (i)compact and diffeomorphic to Snor (ii)smoothly asymptot-
ically conical.
Lemma 10.3 ([11, Proposition 3.5]) Fix n≥3and ≤λn−1and ε>0. Assume that
(n,)and (n, )hold.Then,the space of compact or non-flat smoothly asymp-
totically conical shrinkers ⊂Rn+1with entropy λ() ≤−εis compact14 in
C∞
loc.
From now on, we fix n≥3, ∈(λn,λ
n−1]satisfying (n,)and (n, ).
Lemma 10.4 There is δ=δ(n,,ε) > 0so that if Mis a unit-regular integral
Brakke flow with M(t) =Hn$√−t for t<0, where is a compact or non-
flat smooth asymptotically conical shrinker with F() ≤−ε,then for any X∈
Rn+1×Rwith |X|=1, we have that M(X) ≤F()−δ.
Proof Note that for compact shrinkers one has M(X) ≤1 for all X∈Rn+1×R
with |X|=1 so White’s local regularity theorem [108] yields the statement.
For the noncompact case, assume there is Mj(and the associated smooth asymp-
totically conical shrinkers j) and Xjwith |Xj|=1 so that
Mj(Xj)≥F(
j)−1
j.
14We emphasize that because <2, any limit of such shrinkers has multiplicity one. Note that the proof
of [11, Proposition 3.5] directly allows to include compact shrinkers with λ() ≤−ε.Furthermore,
[11, Proposition 3.7] gives an extrinsic diameter bound for such compact shrinkers, so a sequence of these
cannot converge to a non-compact shrinker.
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Mean curvature flow with generic initial data 179
Up to a subsequence, we can use Lemma 10.3 to find a Brakke flow M∞so that
M∞=Hn$√−t∞for t<0, where ∞is a non-flat smooth asymptotically con-
ical shrinker, and Xwith |X|=1 and M∞(X) ≥F(
∞). Parabolic cone-splitting
(cf. [104] and [29, p. 840-1]) implies that either ∞splits off a line or it is static or
quasi-static. This is a contradiction, completing the proof.
Lemma 10.5 For integral unit regular Brakke flows Mi,M,suppose that Xi∈
singMihas MiMand Xi→X∈sing M.Suppose that some tangent flow to
Mat Xis a round shrinking sphere with multiplicity one,t→Hn$Sn√−2nt.Then,
for isufficiently large,any tangent flow to Miat Xiis a round shrinking sphere.
Proof Assume that X=(0,0). For any r>0, there is η>0, so that
M$(B2r√η(0)×(−4η, −η))
is a smooth, strictly convex mean curvature flow (without spatial boundary). Thus,
for isufficiently large,
Mi$(Br√η(0)×(−3η, −2η))
is a smooth, strictly convex mean curvature flow. Taking rsufficiently large, this
completes the proof (using e.g., [66]).
For any integral unit regular Brakke flow Mwith λ(M)<λ
n−1=λ(Sn−1×R),
denote
singgen M
for the set of singular points for which all tangent flows given by multiplicity-one
round shrinking spheres. The previous lemma proves stability of these sets.
Assume for now that Mn⊂Rn+1has λ(M) < , and consider ϕ∈C∞(M),
ϕ>0. Fix s0small enough so that for s∈(−s0,s
0), the graph of sϕ, denoted Ms,
has λ(Ms)<−ε,forε>0 fixed. For any s∈(−s0,s
0),letF(s ) denote the set
of integral unit regular Brakke flows Mwith M(0)=Hn$Ms. Note that F(s) = ∅
(e.g., choose si→swith the level set flow of Msinon-fattening and pass Brakke
flows starting from Msi—these exist by [74, Theorem 11.4]—to the limit).
For s∈(−s0,s
0),set
D(s) := sup{M(X) :M∈F(s ), X ∈sing M\singgen M}.
We recall that sup ∅=−∞and note that by compactness of integral unit-regular
Brakke flows and upper-semicontinuity of density, D(s ) is always attained.
Proposition 10.6 We continue to assume that λ(M) ≤−ε.For s0as above,suppose
that sis∈(−s0,s
0).Then
lim sup
i→∞ D(si)≤D(s) −δ,
where δ=δ(n,,ε) > 0is fixed in Lemma 10.4.
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180 O. Chodosh et al.
Proof It suffices to assume that limsupi→∞ D(si)>−∞. Choose integral unit-
regular Brakke flows Mi∈F(si)and space-time points Xi∈singMi\singgen Mi
with
Mi(Xi)=D(si).
Passing to a subsequence, we can assume that MiM∈F(s) and
Xi→X∈singM\singgen M.
The fact that X/∈singgen Mfollows from Lemma 10.5. We now rescale around Xso
that we can apply Theorem 9.2. Note that supp Mi, suppMare all pairwise disjoint,
since their initial conditions are compact pairwise disjoint hypersurfaces.
We will repeatedly pass to subsequences without relabeling in the following.
Rescale Miaround Xby |Xi−X|=0to ˆ
Miand assume that ˆ
Miˆ
M. Similarly,
rescale Maround Xby |Xi−X|=0to
Miand assume that
Mi
M. Since
M
is a tangent flow to Mat X/∈singgen M, by Lemma 10.2, there is a smooth shrinker
n⊂Rn+1that is either compact or asymptotically conical so that
M(t) =√−t
for t<0. Finally, assume that after rescaling Xiaround Xby |Xi−X|to ˆ
Xi,
ˆ
Xi→ˆ
X.
We claim that λ( ˆ
M)≤M(X) =F(). Indeed, choose Xi→Xand ri→0so
that
Mi(Xi,r
i)=λ( ˆ
M)+o(1).
On the other hand,
M(X, r ) =M(X) +o(1)
as r→0. Hence,
M(X, r ) =lim
i→∞Mi(Xi,r)≥lim
i→∞Mi(Xi,r
i)=λ( ˆ
M).
Sending r→0 shows that λ( ˆ
M)≤M(X).
Consider a tangent flow to ˆ
Mat −∞. Since λ( ˆ
M)≤, Lemma 10.2 implies
that any such tangent flow is the shrinking flow associated to a smooth shrinker ˆ
.
We claim that ˆ
=and that ˆ
Mlies (weakly) on one side of the shrinking flow
associated to . Indeed, by the Frankel property for self-shrinkers (Corollary D.4),
there is x∈ˆ
∩. Because ,ˆ
have multiplicity one, we can find regions in
ˆ
Mi,˜
Mithat are (after a common rescaling) smooth graphs over connected regions
in ˆ
and containing x. Because supp Miand supp Mare disjoint, it must hold
that ˆ
=. Applying Lemma 7.18 (and the maximum principle), we can find a se-
quence of times ti→−∞so that either ˆ
M(ti)=Hn$√−ti,or ˆ
M(ti)is a smooth
graph over √−tiof a nowhere vanishing function. In the first case, we see that
ˆ
M(t) =Hn$√−t for all t<0 by Proposition C.1 (cf. the proof of Lemma 7.16),
while in the second case, we see that supp ˆ
M(t) is disjoint from √−t for all t<0
(by Ilmanen’s localized avoidance and the Ecker–Huisken maximum principle, as in
Lemma 7.15).
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Mean curvature flow with generic initial data 181
We claim that the second case cannot occur. Indeed, Theorems 9.1 and 9.2 (and
≤λn−1) imply (since λ( ˆ
M)≤F()) that sing ˆ
M=singgen ˆ
M, so Lemma 10.5
implies that ˆ
Xi∈singgen ˆ
Mifor isufficiently large. This is a contradiction, so the
first case (i.e., ˆ
M(t) =√−t for t<0) must hold.
Now, we can apply Lemma 10.4 to ˆ
Mand ˆ
X(the limit of the rescaled points ˆ
Xi;
note that |ˆ
X|=1) to conclude that
lim sup
i→∞ D(si)=lim sup
i→∞ Mi(Xi)≤ˆ
M(ˆ
X) ≤F()−δ≤D(s) −δ.
This completes the proof.
Using this, we can prove the existence of generic flows.
Theorem 10.7 For n≥3and ∈(λn,λ
n−1],assume that (n,)and (n, )hold.
Then if M⊂Rn+1is a closed hypersurface with λ(M) < and ϕ>0is a smooth
positive function on M,let Msdenote the normal graph of sϕ over M.Then,there
is s0>0and a closed,countable set B⊂(−s0,s
0)so that for s∈(−s0,s
0)\B,any
unit-regular integral Brakke flow with initial condition Msis completely regular until
it disappears in a round point.
Proof Note that
B={s∈(−s0,s
0):D(s) > −∞}
is closed by upper semicontinuity of density and Lemma 10.5. Thus, it suffices to
prove that Bis countable. Define
Bj:={s∈(−s0,s
0):D(s) ∈[−jδ,−(j +1)δ)},
so B=∪J
j=0Bj,forJ>
δ. By Proposition 10.6,ifs∈Bj, there is an open interval
Iso that I∩Bj={s}. Hence, Bjis countable. This completes the proof.
Proof of Theorem 10.1 By [42, Theorem 4.30], if Mis not a round sphere, then after
replacing Mby a nearby C∞-close hypersurface, we can assume that λ(M) ≤−ε
for some ε>0. The statement then follows from Theorem 10.7.
Corollary 10.8 For n≥3and ∈(λn,λ
n−1],assume that (n,)and (n, )hold.
Consider the set
Emb<(Rn+1):= {M⊂Rn+1,λ(M)< }
with the C∞topology.15 Define a subset Gby the set of M∈Emb<(Rn+1)so that
any unit-regular integral Brakke flow starting from Mis regular until it disappears
in a round point.Then Gis open and dense.
15For example, we can say that Mj→Mif for jlarge, Mj=graphMujwith uj→0inC∞(M).
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182 O. Chodosh et al.
Proof We claim that Emb≤(Rn+1)\Gclosed. Consider Mj∈Emb≤(Rn+1)\G
with Mj→M∈Emb≤(Rn+1).LetMjdenote integral unit-regular Brakke flows
starting at Mjwith non-round tangent flows at Xj. Passing to a subsequence, Mj
M, a integral unit-regular Brakke flow starting from M. By Lemma 10.5, a further
subsequence has Xj→X∈sing Mwith Mhaving a non-round tangent flow at X.
This shows Gis open. Finally, the density of Gfollows from Theorem 10.7.
11 The first non-generic time for flows in R3
In this section, we will study the mean curvature flow of a generic initial surface in
R3. We will remove the low-entropy assumption considered in the previous section
and study the possible singularities that generically arise.
For Man integral unit-regular Brakke flow, define Tgen to be the supremum of
times Tso that at any point X∈supp Mwith t(X) < T , all tangent flows at Xare
multiplicity-one spheres, cylinders, or planes.
Theorem 11.1 Suppose that M⊂R3is a closed embedded surface of genus g.Then,
there exist arbitrarily small C∞graphs Mover Mand corresponding cyclic integral
unit-regular Brakke flows Mwith M(0)=H2$M,so that either:
(1) Tgen(M)=∞,or
(2) there is x∈R3so that some tangent flow to Mat (x,T
gen(M)) is kH2$√−t
for a smooth shrinker of genus at most gand either:k≥2or has a cylin-
drical end but is not a cylinder.
We will prove this below. Note that Theorem 11.1 yields the following conditional
result. Recall that the list of lowest entropy shrinkers is known to be the plane, the
sphere, and then the cylinder by [8,46,108]. Suppose that there is g∈(λ1,2]so
that any smooth shrinker ⊂R3with genus() ≤gand F() <
gis either a
plane, a sphere, a cylinder, or has no cylindrical ends.16 Then:
Corollary 11.2 If M⊂R3is a closed embedded surface with genus(M) ≤gand
λ(M) ≤g,then there are arbitrarily small C∞graphs Mover Mand cyclic
integral unit-regular Brakke flows Mwith M(0)=H2$Mso that Mhas only
multiplicity-one spherical or cylindrical tangent flows,i.e., Tgen(M)=∞.
We now establish certain preliminary results used in the proof of Theorem 11.1.
The proof of Theorem 11.1 can be found after the statement of Proposition 11.4.We
define
singgen(M)⊂sing(M)
16Work of Brendle [20] implies that only possible genus zero self-shrinkers are the plane, sphere, and
cylinder. This immediately implies that 0=2. Ilmanen has conjectured that no non-cylindrical shrinker
can have cylindrical ends [78, #12], which would mean we can take g=2forallg. However, it could
theoretically happen that the next lowest entropy shrinker is a counterexample to Ilmanen’s conjecture,
i.e., has a cylindrical end.
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Mean curvature flow with generic initial data 183
as the set of singular points so that one tangent flow (and thus all of them by [47];
alternatively, this follows from [44,91]or[11]) is a multiplicity-one shrinking sphere
or cylinder.
First, we note the following result establishing regularity of tangent flows at Tgen
(see also the proof of [39, Theorem 1.2]).
Proposition 11.3 Consider a cyclic integral unit-regular Brakke flow Min R3,with
M(0)=H2$Mfor a closed embedded surface M.Then,Mhas the following prop-
erties:
•the level set flow of Mdoes not fatten before Tgen(M),
•for almost every t∈[0,T
gen(M)],the level set flow of Mis given by M(t),a
smooth embedded surface,
•M(t) =H2$M(t) for almost every t∈[0,T
gen(M)],and
•t→genus(M(t )) is non-increasing for all smooth times t∈[0,T
gen(M)].
Furthermore,assuming that Tgen(M)<∞,then any tangent flow ˆ
Mat (x,T
gen(M))
satisfies
ˆ
M(t) =kH2$√−tˆ
for t<0,
where ˆ
is a smooth embedded self-shrinker with
genus(ˆ
) ≤lim
tTgen(M)
t/∈t−1(sing M)
genus(M(t)).
Moreover,ˆ
has finitely many ends,each of which is either asymptotically conical or
cylindrical (with multiplicity one), and if (x,T
gen(M)) ∈sing(M)\singgen(M)and
k=1, then ˆ
has genus(ˆ
) ≥1.
Proof By [39, Theorem 1.9], the level set flow of Mdoes not fatten for t∈
[0,T
gen(M)). Hence, by [45, Corollary 1.4] and Corollary G.5, for almost every
t∈[0,T
gen(M)], the level set flow of Mat time tis a smooth surface M(t) and we
have that M(t) =H2$M(t).Now,by[103, Theorem 1] (cf. [110]) t→genus(M(t ))
is non-increasing.
These two facts suffice to repeat the proof in [76] with only superficial changes
(to avoid the singular time-slices) Since this is a crucial point, we describe these
modifications in some more detail here. The monotonicity formula gives that along
the blow-up sequence corresponding to the given tangent flow, we have
−1
−1−τλj(M(T +λ2
jt)−x)∩Br(0)|H|2dμ(t)dt ≤Cτr2R2+δR(λj)
as in [76, §3], where Cdepends only on an entropy bound for M,r<Rare fixed,
and δR(λj)→0asj→∞. Note that in [76, §3], this is proven for a smooth flow,
but the same proof holds here since the flow is smooth for a.e. time and this inequality
is valid at the level of a Brakke flow. Thus, since a.e. time tis a regular time, we can
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184 O. Chodosh et al.
follow [76, Proof of Theorem 1] almost verbatim, except we can insist that tj→−1
is chosen so that T+λ2
jtjis a smooth time for M(t).By[76, Theorem 3], λj(M(T +
λ2
jtj)−x)has second fundamental form uniformly bounded in L2on compact sets.
The remainder of the proof that ˆ
M(t) =kH2$√−tˆ
for ˆ
smooth embedded self-
shrinker is then completed exactly as in [76, Proof of Theorem 1].
This proves all but the last two claims. Finally, the statement about the ends of ˆ
isprovenin[102](cf.[99, Appendix A]), while genus zero shrinkers are classified in
[20].
We note that by Proposition 11.3, we can unambiguously define:
genusTgen (M):= lim
tTgen(M)
t/∈t−1(sing M)
genus(M(t)),
the genus of Mright before the first non-generic singular time. This notion will be
useful in the following proposition which will be the key mechanism used to perturb
away asymptotically conical (and compact, non-spherical) singularities.
Proposition 11.4 Suppose that M⊂R3is a closed embedded surface of genus gand
Mis a cyclic integral unit-regular Brakke flow with M(0)=H2$M.Assume that
Tgen(M)<∞and that any tangent flow at time Tgen(M)has multiplicity one and
that there is no non-cylindrical tangent flow at time Tgen(M)with a cylindrical end.
Then,there exists arbitrarily small C∞graphs Mover M,and cyclic integral
unit-regular Brakke flows Mwith M(0)=H2$M,so that
Tgen(M)>T
gen(M)and genusTgen (M)≤genusTgen (M)−1.
Before proving this, we will show that it implies the full genericity result.
Proposition 11.4 implies Theorem 11.1 For Ma closed embedded surface of genus g,
consider any cyclic integral unit-regular Brakke flow Mwith M(0)=H2$M. Such
aflowMexists by [89, Theorem B.3] (alternatively, one could perturb Mslightly at
this step so that the level set flow of Mdoes not fatten, and apply [74, Theorem 11.4]).
First, suppose that either Tgen(M)=∞or Tgen (M)<∞but at Tgen(M)there
is a tangent flow that either has multiplicity greater than one or is a non-cylindrical
shrinker with a cylindrical end. In this case, we can take M=Mand M=M,
completing the proof. In case this does not hold, Proposition 11.4 yields a small C∞
perturbation M1of M, and a Brakke flow M1with M1(0)=H2$M1. Moreover,
genusTgen (M1)≤genusTgen (M)−1≤genus(M) −1.
At this point, we can iterate. Either M1satisfies the desired conditions, or Proposi-
tion 11.4 applies to M1. In the former case, we can conclude the proof, and in the
latter case we find a small C∞perturbation M2of Mwith a Brakke flow M2as
above. Repeating this process ktimes, we find that
genusTgen (Mk)≤genus(M) −k.
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Mean curvature flow with generic initial data 185
By Proposition 11.3, it must eventually hold that Mk,Mksatisfies one of the two
desired conclusions (1) or (2) for some k≤genus(M). Thus, after at most genus(M)
perturbations, we find the desired M=Mkand M=Mk. This completes the proof.
The proof Proposition 11.4, will depend on the following lemmata.
Lemma 11.5 There is δ0>0so that if Mis a cyclic integral unit-regular Brakke flow
in R3with M(0)=H2$Mfor a smooth surface M,then for any
X∈(sing(M)\singgen(M)) ∩{t=Tgen(M)},
we have M(X) ≥λ(S1)+δ0.
Proof This follows by combining Proposition 11.3 with [8, Corollary 1.2].
Lemma 11.6 (cf. [11, Theorem 4.3]) Suppose that Mis a cyclic integral unit-regular
Brakke flow in R3with M(0)=H2$Mfor some closed embedded surface M.As-
sume that Tgen(M)<∞and that any tangent flow at time Tgen(M)has multiplicity
one and that there is no non-cylindrical tangent flow with a cylindrical end.17
Then for (x0,T
gen(M)) ∈sing(M)\singgen(M),there are r, ρ, τ > 0so that
M1:=M$(B4r(x0)×(Tgen(M)−2τ,Tgen(M)]\{(x0,T
gen(M))},
M2:=M$((B4r(x0)\Br/4(x0)) ×(Tgen(M)−2τ, Tgen(M)+2τ)
are smooth mean curvature flows.Moreover,any (x,t)∈supp M∩(U1∪U2)satisfies
|(x−x0)⊥|≤ 1
10 |x−x0|,(11.1)
where
U1:={(x0+x,T
gen(M)−t) :0<ρ
2t<|x|2<16r2,t <2τ},
U2:=(B4r(x0)\Br/4(x0)) ×(Tgen(M)−2τ, Tgen(M)+2τ).
Proof Observe that by Proposition 11.3 and the given hypothesis, any18 tangent flow
at (x0,T
gen(M)) is associated to a smooth multiplicity-one shrinker that is either
compact or asymptotically conical.
We begin by proving that the smoothness assertion holds for M1for any r, τ >
0 small. Indeed, suppose there are singular points Xi:= (xi,T
gen(M)−ti)→
(x0,T
gen(M)) with ti≥0, rescaling around (x0,T
gen(M)) to ensure that Xiare a
unit distance from the space-time origin, we would find a singular point in a tangent
flow to Mat (x0,T
gen(M)) lying in the parabolic hemisphere
{(x,t):t≤0,|x|2+|t|=1}.
17That is, assume that we cannot simply take M=Min Theorem 11.1.
18We emphasize that while we do not need to refer to uniqueness of the tangent flow in this proof, it does
indeed hold in this setting by [91] for compact tangent flows and [31] for asymptotically conical ones.
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186 O. Chodosh et al.
However, no such point in the tangent flow can be singular (since such a flow would
not be asymptotically conical).
We now consider (11.1) for points in suppM∩U1. Note that by the smoothness
of M1, all such points are smooth points of M. We claim that there is ρ>0 suffi-
ciently large so that (11.1) holds in suppM∩U1, after shrinking r, τ > 0 if necessary.
Choose (x0+x,T
gen(M)−t) ∈supp Mwith (x,t)→(0,0)and 0 <ρ
2t<|x|2but
so that
|x⊥|≥ 1
10 |x|.
Rescaling around (x0,T
gen(M)) and passing to the limit, we find a tangent flow to M
at (x0,T
gen(M)) with associated shrinker ρso that for some xρ∈with |xρ|≥ρ
|x⊥
ρ|≥ 1
10 |xρ|.
However, this will be in contradiction to [43], Proposition 11.3, and the fact that the
set of tangent flows is compact.19 Indeed, consider Brakke flows Mρassociated to
ρ. We consider the point (ρ−1xρ,−ρ−2)and take a subsequential limit of Mρto
find ˜
Ma shrinking flow associated to ˜
an asymptotically conical shrinker; how-
ever, the subsequential limit (˜
x,0)of the space-time points (ρ−1xρ,−ρ−2)lies on
the asymptotic cone of ˜
(and is not at the origin) and thus has ˜
x⊥=0. This is a
contradiction, completing the proof.
Finally, we prove both the smoothness of M2and (11.1) for points in supp M∩
U2. If some tangent flow to Mat (x0,T
gen(M)) is compact, then by considering
shrinking spherical barriers, we can choose r, τ > 0 so that M2is empty. As such,
we can assume that there is an asymptotically conical shrinker associated to some
tangent flow Mof Mat (x0,T
gen(M)). Because is asymptotically conical,
|x||A(x)|=O(1)and |x⊥|≤o(1)|x|as x→∞. Arguing as in [31, Lemma 9.1],
we can use pseudocality (e.g., [80, Theorem 1.5]) on large balls along the end of
to find R>0 sufficiently large so that
M$((Rn+1\BR)×[−1,1])is smooth
and satisfies |x⊥|≤ 1
100 |x|. From this, we can choose r, τ > 0 so that the assertions
about M2follow after choosing a blow-up sequence at (x0,T
gen(M)) converging
to M.
Proof of Proposition 11.4 Fix a closed set Kwith ∂K =M. Choose smooth surfaces
Mi=∂Kiwith Miconverging to Min C∞, where Kiare closed sets with Ki⊂Ki+1
and Mi∩Mi+1=∅and Mi∩M=∅. We can moreover assume that the level set flow
of Midoes not fatten [74,p.63],soby[74, Theorem 11.4] there is a cyclic integral
unit-regular Brakke flow Miwith Mi(0)=H2$Mi.
Passing to a subsequence, Miconverges to a Brakke flow M∞with M∞(0)=
H2$M. On the other hand, combining Proposition 11.3 with Corollary G.5, we find
19Alternatively, one may argue as follows: by [31], ρis independent of ρ, which immediately yields a
contradiction since for any fixed asymptotically conical shrinker, |x⊥|≤o(1)|x|as x→∞.
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Mean curvature flow with generic initial data 187
that M∞=Mfor t∈[0,T
gen(M)). In particular, Tgen(M)=Tgen(M∞)and any
tangent flow to M∞at time Tgen(M∞)has multiplicity one and no such tangent
flow is non-cylindrical but with a cylindrical end.
We claim that for isufficiently large, M=Miand M=Misatisfy the assertion.
Note that Lemma 11.5 and upper-semicontinuity of density imply that
lim inf
i→∞ Tgen(Mi)≥Tgen(M∞).
We claim that
Tgen(Mi)>T
gen(M∞)
for sufficiently large i. If not, we can pass to a subsequence so that
Tgen(Mi)≤Tgen(M∞). (11.2)
We claim that this leads to a contradiction using the strategy of proof from Proposi-
tion 10.6. Choose
Xi∈(sing(Mi)\singgen(Mi)) ∩{t=Tgen(Mi)},
and let Xi→X∞.By(11.2) and Proposition 11.3 any tangent flow to M∞at X∞is
associated to a multiplicity one smooth shrinker with all ends (if any) asymptotically
conical (note that X∞cannot have a multiplicity-one cylindrical or spherical tangent
flow by Lemma 11.5). In particular, Theorems 9.1 and 9.2 apply to the shrinkers as-
sociated to any tangent flow to M∞at X∞. We now use these results to obtain a
contradiction to (11.2). Briefly, the strategy will be as follows: rescaling Xiaround
X∞we obtain a flow that lies weakly to one-side of a self-shrinking tangent flow to
M∞. If the flow lies strictly to one side, it has no non-spherical/cylindrical singu-
larities so we obtain a contradiction. On the other hand, if it agrees with the tangent
flow for t<0 then we use the observation that a conical or compact shrinking flow
is smooth up to and including t=0 except at the origin. (Note that this would fail if
the shrinker had a cylindrical end.) This will contradict (11.2).
We now give the full argument. After rescaling by |Xi−X∞|=0, the flows Mi
converge either to a flow on one side of the tangent flow to M∞at X∞or a flow
which agrees with a tangent flow to M∞for t<0. In the first case, the limit has
only multiplicity one cylindrical and spherical singularities by Theorems 9.1 and 9.2.
This contradicts the choice of Xiby Lemma 11.5. On the other hand, the second
case cannot occur. Indeed, if the second case occured, then (11.2) would imply that
some tangent flow to M∞has a singularity at (x,t) with |(x,t)|=1 and t≤0,
contradicting Proposition 11.3 and the assumption that no non-cylindrical tangent
flow to M∞at Tgen(M∞)has cylindrical ends.
As such, since the flows Miare converging to a flow on one-side of the tangent
flow to M∞at X∞, we see that
Tgen(Mi)>T
gen(M∞)(11.3)
for isufficiently large by (8) in Theorem 9.1 combined with Theorem 9.2.
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188 O. Chodosh et al.
It remains to prove the strict genus reduction. As above, we first briefly sketch the
idea for the reader’s convenience. By the work of Brendle [20], every non-generic
singularity that occurs at time Tgen(M)has to have positive genus. Lemma 11.6
will be used to show that this positive genus is captured in the tangent flow scale of
our non-generic singularities. Our understanding of the long-time behavior of flows
to one side of a non-generic shrinker (Theorems 9.1,9.2) and Lemma 11.6 again
will then imply that, near the non-generic singularities of M∞, the one-sided flows
Miwill experience strict genus reduction. The result will follow by a localization
of the well-known genus monotonicity property of mean curvature flow, given in
Appendix H.
We assume that
(0,T
gen(M∞)) ∈sing(M∞)\singgen(M∞).
Fix the corresponding parameters r,ρ,τas in Lemma 11.6.
Define20
di:=d(suppMi(Tgen(M∞)), 0)>0.
Note that limi→∞ di=0. Moreover, rescaling M∞(resp. Mi) around
(0,T
gen(M∞)) by dito M∞,i (resp. ˜
Mi), we can pass to a subsequence so that
as i→∞,M∞,i converges to a tangent flow to M∞at (0,T
gen(M∞)) and ˜
Mi
converges to a parabolic dilation of the ancient one-sided flow described in Theo-
rems 9.1 and 9.2 associated to this tangent flow.
We begin by proving the following two claims that imply that perturbed flows Mi
lose genus locally around points x.
Claim (A) There is ¯τ∈(0,τ]so that for any isufficiently large and t∈[¯τ,2¯τ],
Mi(Tgen(M∞)−t)$B3r(0)
is smooth,21 intersects ∂B2r(0)transversely,and Mi(Tgen (M∞)−t)$B2r(0)has
positive genus.22
Claim (B) For isufficiently large,there is ¯ε=¯ε(i) > 0so that for t∈[0,¯ε)
Mi(Tgen(M∞)−t)$B3r(0)
is a smooth genus zero surface.
20Note that this is spatial (Euclidean) distance.
21The restriction to a ball Bof a time-tslice of a Brakke flow M, i.e., M(t)$B,issaidtobesmooth
if t−1(t) ∩sing M∩B=∅. Note that this is stronger than simply asserting M(t )$B=H2$Mfor some
smooth surface M⊂B. For example, the flow associated to a shrinking sphere disappearing at time T
satisfies M(T ) =H2$∅,butM(T ) is not smooth in the sense above.
22Recall: the genus of a surface (possibly with boundary) properly embedded in a ball B⊂R3is the
genus of the surface obtained from after capping off each boundary component with a disk.
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Mean curvature flow with generic initial data 189
Proof Claim (A) By Proposition 11.3, any tangent flow to M∞at (0,T
gen(M∞)) has
multiplicity one and positive genus. Thus, by Lemma 11.6,23 we can take ¯τsuffi-
ciently small so that M∞(T∞(M∞)−t)$B4r(0)is smooth and has positive genus
for t∈[¯τ/2,3¯τ]. Combined with Brakke’s theorem [108] and another application of
Lemma 11.624 the remaining assertions follow.
Proof of Claim (B) We have fixed a tangent flow to M∞and associated one-sided flow
from Theorem 9.1.Letδ>0 denote the interval of regularity around t=0forthe
one-sided flow, as described in property (10) of Theorem 9.1. We thus define
¯ε(i) =δd2
i
2.
This will ensure that when rescaling by di, we are considering a short enough time
interval to apply (10) in Theorem 9.1.
We first show that for isufficiently large, Mi(Tgen(M∞)−t)$B3r(0)is smooth
for all t∈[0,¯ε(i)). Suppose, instead, that there were some yi,tisuch that
yi∈(singMi(Tgen (M∞)−ti)) ∩B3r(0), ti∈[0,¯ε(i)). (11.4)
Since
Mi$(B4r(0)×{t≤Tgen(M∞)})M∞$(B4r(0)×{t≤Tgen(M∞)})
as Brakke flows (for i→∞), it follows by Lemma 11.625 that yi→0as i→∞.
On the other hand, by definition of ¯ε(i) and (10) in Theorem 9.1,
˜
di:=d((yi,T
gen(M∞)−ti), (0,T
gen(M∞)) di.(11.5)
In particular, rescaling Miby ˜
diaround (0,T
gen(M∞)), the flow converges to some
flow ˜
M∞.By(11.5), we have that (0,0)∈supp ˜
M∞. Thus, we have that for t<0,
˜
M∞agrees with a tangent flow to M∞at (0,T
gen(M∞)). This is a contradiction
since Proposition 11.3 implies that ˜
M∞$((Rn+1×(−∞,0])\{(0,0)})is smooth.
Thus, no points yias in (11.4) will exist. This completes the proof of the regularity
assertion.
We finally prove that for ti∈[0,¯ε(i)), the surface Mi(Tgen(M∞)−ti)$B3r(0)
has genus zero for ilarge. We show below that for some R>0 sufficiently large
(independent of i), for any ilarge and
x∈suppMi(Tgen (M∞)−ti)∩(B3r(0)\BRdi(0)), (11.6)
we have |x⊥|≤1
5|x|. This follows from essentially the same scaling argument as
above. Indeed, consider a sequence of a points yiand times tiviolating this bound
23Specifically, the regularity of M1.
24Specifically, (11.1) on supp M∩U1.
25Specifically, the smoothness of M1.
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190 O. Chodosh et al.
while still satisfying (11.6) (we will choose R>0 large below). Rescaling Mi
around (0,T
gen(M∞)) by
˜
di:=d((yi,T
gen(M∞)−ti), (0,T
gen(M∞)),
we claim that it now must hold that
lim sup
i→∞
˜
di
di
<∞.(11.7)
Indeed, if this fails, we can argue precisely as in the previous paragraph to rescale by
˜
dito find a tangent flow to M∞at (0,T
gen(M∞)); the points (yi,t
i)converge–after
rescaling–to a point on the tangent flow at t=0 (at a unit distance from 0). Clearly
the cone satisfies the asserted bound, so this is a contradiction.
Thus, (11.7) holds. In particular, the points (yi,−ti)remain a bounded distance
from (0,0)when rescaling by di(but lie outside of BR(0)×R). It is easy to see26
that we can take R>0 large so that the one-sided flow from Theorem 9.1 (scaled
to have unit distance from (0,0)) satisfies |x⊥|≤ 1
10 |x|for (x,t) with |x|≥Rand
|t|<δ.
We now demonstrate that putting these facts together, we have proven the claim.
After rescaling by dithe flows Miconverge to the one sided flow to the tangent flow
of M∞. By property (10) of Theorem 9.1 this one sided flow at time zero is smooth
and strictly star-shaped and therefore has genus zero. Thus by smooth convergence
and thus the transverse intersection at the boundary of the ball B2Rdi(0), together with
the choice of ¯ε(i),Mi(Tgen(M∞)−ti)$B2Rdi(0)has genus zero for ilarge.
Now, take ¯τsmaller if necessary and then fix ilarge. We write Mi=Mand
assemble the following properties established above:
(1) For t∈(Tgen(M∞)−2¯τ,T
gen(M∞)+2¯τ) a smooth time for Mwe have that
M(t) =H2$M(t),
for M(t) smooth with
genus(M(t)) ≤genusTgen (M∞);
this follows from the monotonicity of genus (cf. Proposition 11.3) and the fact
that MiM∞as Brakke flows.
(2) For t∈[¯τ,2¯τ],M(T∞(M∞)−t)$B3r(0)is smooth and has positive genus in
B2r(0); this was proven in Claim (A) above.
(3) There is 0 <¯ε< ¯τso that for t∈[0,¯ε),M(T∞(M∞)−t)$B3r(0)is smooth
and has zero genus; this was proven in Claim (B) above.
(4) We have that
M$((B3r(0)\Br/2(0)) ×(Tgen(M∞)−2¯τ,T
gen(M∞)+2¯τ))
26By the argument in Lemma 7.16, the blow-down of the ancient one-sided flow agrees for t<0 with the
shrinking Brakke flow associated to the fixed shrinker.
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Mean curvature flow with generic initial data 191
is a smooth flow of (a disjoint union of) topological annuli, intersecting ∂Br(0)
transversely for r≤r≤3r; this follows from Lemma 11.6 and the fact that
MiM∞as Brakke flows.
Choose
¯
t1∈(Tgen(M∞)−2¯τ,T
gen(M∞)−¯τ],
¯
t2∈(Tgen(M∞)−¯ε,Tgen(M∞)]
smooth times for M. We claim that
g:=genus(M(¯
t2)) < genus(M(¯
t1)). (11.8)
By property (1) in the above list (and monotonicity of genus, cf. Proposition 11.3),
once we have established (11.8), we will find
genusTgen (M)≤genus(M(¯
t2)) ≤genusTgen (M∞)−1,
which will complete the proof.
It thus remains to establish (11.8). We will show this by combining the properties
above with a localization of White’s [103] topological monotonicity, which we have
included in Appendix H. Define
B:=B2r(0).
The key observation, which makes Appendix Happlicable, is that, by property (4)
above, the level set flow for times in [¯
t1,¯
t2]of M(¯
t1)×{¯
t1}(which must agree with
the restriction of M)isasimple flow (defined in Appendix H) in the tubular neigh-
borhood
U:=B3r(0)\¯
Br(0),
of ∂B for t∈[¯
t1,¯
t2]. We can thus apply results of that appendix with [¯
t1,¯
t2]in place
of [0,T], and R3\¯
Bin place of . (Certainly, we can and will also apply White’s
global topological monotonicity results.) We invite the reader to recall the notation
W[¯
t1,¯
t2],W[¯
t1],W[¯
t2]from (H.1)-(H.2) in Appendix H, which we’re going to make
use of here.
To quantify the genus drop, we’ll use Lemmas 11.7 and 11.8 stated and proved be-
low. Loosely speaking, Lemma 11.7 constructs a good choice of linearly independent
set of loops in H1(W [¯
t2])detecting the number gand compatible with the geome-
try (namely, the smoothness of the flow in the annular region as established in (4)
above). By the localized version of White’s topological monotonicity established in
Appendix H, we can homotop these loops back to time ¯
t1. The properties established
in Lemma 11.7 are preserved under this process and then we can apply Lemma 11.8
to show that the genus at time ¯
t1wouldhavetobe≤g. This proves the desired genus
monotonicity.
Choose loops γ¯
t2
1,...,γ¯
t2
2gin W[¯
t2]as in Lemma 11.7. That is,
{[γ¯
t2
1],...,[γ¯
t2
2g]}⊂ H1(W [¯
t2])
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192 O. Chodosh et al.
is linearly independent and each γ¯
t2
isatisfies either:
•γ¯
t2
iis contained in ¯
Bc(since genus(M(¯
t2)∩B) =0 implies that no γ¯
t2
ican be
contained in B), or
•there is some component Ui[¯
t2]of W[¯
t2]∩∂B that has non-zero signed intersection
with γ¯
t2
i, and zero signed intersection with each previous γ¯
t2
j,j<i.
By the injectivity of H1(W [¯
t2])→H1(W [¯
t1,¯
t2])[103, Theorem 6.2], the inclusion
{[γ¯
t2
1],...,[γ¯
t2
2g]}⊂ H1(W [¯
t1,¯
t2])
is linearly independent too. We now construct loops γ¯
t1
1,...,γ¯
t1
2gin W[¯
t1]so that:
•Each γ¯
t2
iis homotopic to γ¯
t1
iin W[¯
t1,¯
t2];see[103, Theorem 5.4].
•If γ¯
t2
iis entirely contained in ¯
Bc, then so is γ¯
t1
iand the entire homotopy between
them; see Theorem H.3.
•If γ¯
t1
iis not entirely contained in ¯
Bc, there is some component Ui[¯
t1]of W[¯
t1]∩∂B
that has non-zero signed intersection with γ¯
t1
i, and zero signed intersection with
each previous γ¯
t1
j,j<i; this follows from the simplicity of the flow in U×[¯
t1,¯
t2]
and the fact that signed intersection is preserved under homotopy.
We can now easily complete the proof. If (11.8) were false, then genus(M(¯
t1)) =
gby White’s global topological monotonicity [103]. Applying Lemma 11.8 to
γ¯
t1
1,...,γ¯
t1
2gnow says that, because genus(M(¯
t1)∩B) > 0 by property (2) above,
at least one of the γ¯
t1
imust be contained in B, a contradiction.
Lemma 11.7 Suppose that S⊂R3is a closed and embedded genus-gsurface which
is transverse to a sphere ∂B ⊂R3.Denote W:=R3\S.
We can find loops γ1,...,γ
2ginside Wso that {[γ1],...,[γ2g]}⊂ H1(W ) ≈Z2g
is linearly independent and so that,for every i=1,...,2g,either:
•γiis contained in Bor in ¯
Bc,or
•there is a component Uiof ∂B \Sthat has non-zero signed intersection with γi,
and zero signed intersection with each previous γj,j<i.
Moreover,we can arrange that exactly 2genus(S ∩B) of the γiare contained entirely
in Band that if,in H1(W ∩¯
B),
{i:γi⊂B}
ni[γi]=[β]
for some cycle β⊂W∩∂B,then all of the coefficients nivanish.
Proof We induct on the number of components bof S∩∂B.
First, consider b=0. In this case, Sdecomposes into the disjoint union of two
closed surfaces, SB:= S∩B,S¯
Bc:= S\¯
B, which do not meet ∂B.Wehave
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Mean curvature flow with generic initial data 193
genus(SB)+genus(S ¯
Bc)=g, so by applying Alexander duality we find a linearly
independent set
{[γ1],...,[γ2g]}⊂ H1(W )
with 2genus(SB)of the γicontained in B, and the remaining 2 genus(S ¯
Bc)contained
in ¯
Bc. Moreover, W∩∂B =∂B when b=0. Therefore, if a linear combination of
γi⊂Bis homologous to a cycle in W∩∂B then the combination must be =0∈
H1(W ) (since H1(∂B ) =0). This completes the base case.
Now, we consider the inductive step. Consider the bcomponents of S∩∂B.By
the Jordan curve theorem, each component of S∩∂B divides ∂B into two regions. As
such, we can find a component αof S∩∂B so that there is a disk D⊂∂B with ∂D =
αand S∩D◦=∅. Form the surface Sby removing an annulus A=Uε/10(α) ⊂S
and then by gluing two disks that are small deformations of D, into and out of B
respectively, to cap off the boundary of S\A. We can arrange that this all occurs in
Uε(D) ⊂R3(with ε>0 small enough so that Uε(D) is contractible).
The surface Snow satisfies the inductive hypothesis, since S∩∂B has b−1
components. Note that, by definition,
genus(S∩B) =genus(S ∩B), (11.9)
although the genus of Smight be different from Sas we will see below.
There are two cases to consider: either αseparates the component of Sthat con-
tains it, or it doesn’t separate it.
Separating case. Suppose that αseparates the component of Sthat contains it. It
will be convenient to give a name to this component, so let us denote it Sα.Inthis
case, Sα\Ais a disconnected surface with boundary. Hence,27
genus(Sα)=genus(Sα\A),
so genus(S)=g. Applying the inductive step to S(which has b−1<b boundary
circles), we find a linearly independent set of loops γ
1,...,γ
2gin R3\Ssatisfying
the conditions of the lemma with Sin place of S. The curves γ
ithat are not contained
in Bor in ¯
Bchave associated components U
i⊂∂B \Swith the required signed
intersection properties, per the inductive step.
Note that we can assume that the loops γ
1,...,γ
2gare disjoint from Uε(D).As
such, they lie in W, so to prove the inductive step we can simply take
γ1:=γ
1,...,γ
2g:=γ
2g.
For any γithat is not contained in Bor in ¯
Bc,wesetUi:= U
i\Dor Ui:= U
i
depending on whether D⊂U
ior not (respectively). We claim this configuration of
27This can be seen by the inclusion-exclusion principle for Euler characteristic: if we can decompose
a connected surface into two connected components M=M1∪M2where M1and M2intersect in a
circle, then χ(M) +χ(S1)=χ(M1)+χ(M2).Wehavethatχ(S1)=0, χ(M ) =2−2genus(M ),and
χ(Mi)=1−2 genus(Mi)(because they both have a single boundary component). Hence, genus(M) =
genus(M1)+genus(M2).
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194 O. Chodosh et al.
γ1,...,γ
2gsatisfies the properties we want. Note that the two bullet points are just a
consequence of how our curves are disjoint from Uε(D), and that 2 genus(S ∩B) of
the γiare contained in Bin view of (11.9) and the inductive step. It remains to check
two required homological properties.
Suppose there are niso that
{i:γi⊂B}
ni[γi]=[β]in H1(W ∩¯
B),
for some cycle β⊂∂B\S. Note that the components β of βthat intersect Dmust be
fully contained inside D. We write β=β+β. Note further that we can assume that
βconsists of finitely many disjoint embedded circles. Thus, we can find a 2-chain
σ⊂B\Ssuch that
∂σ =β−
{i:γi⊂B}
niγi.
Using the structure of β we see that we can replace σby σ+σ such that σis
contained in ¯
B\(S ∪Uε(D)) ⊂¯
B\Sand σ is contained in ¯
B∩Uε(D). Since the
latter region is contractible (Dwas contractible), this implies that
{i:γi⊂B}
ni[γi]=[β]in H1(¯
B\S).
By the inductive step, all of the coefficients vanish.
We finally show {[γ1],...,[γ2g]}⊂H1(W ) is linearly independent. Assume
n1[γ1]+···+n2g[γ2g]=0inH1(W ). (11.10)
By construction and the inductive step, for any γinot contained entirely in Bor ¯
Bc,
there is a component U
i⊂∂B \Sthat has non-zero signed intersection with γiand
zero signed intersection with each previous γj,j<i. Proceeding from large indices
to small this implies that any γinot contained entirely in Bor in ¯
Bchas ni=0in
(11.10). The Mayer–Vietoris sequence for (W ∩¯
B,W ∩Bc)yields the exact sequence
···→H1(W ∩∂B) →H1(W ∩¯
B) ⊕H1(W ∩Bc)→H1(W ) →···.
Let IBdenote the indices iso that γi⊂Band similarly for I¯
Bc. Consider
⎛
⎝
i∈IB
ni[γi],−
i∈I¯
Bc
ni[γi]⎞
⎠∈H1(W ∩¯
B) ⊕H1(W ∩Bc).
Seeing as we’re assuming this is sent to 0 ∈H1(W ), exactness yields a [β]∈H1(W ∩
∂B) so that
[β]=
i∈IB
ni[γi]in H1(W ∩¯
B), and
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Mean curvature flow with generic initial data 195
[β]=−
i∈I¯
Bc
ni[γi]in H1(W ∩Bc).
We have already seen above, though, that ni=0 for all i∈IBsince βis a cycle in
∂B \S. Thus [β]=0inH1(W ∩¯
B). Arguing as above we can replace βby β(which
has no component in D), such that
[β]=0inH1(¯
B\S)
and
[β]=−
i∈I¯
Bc
ni[γi]in H1(R3\(B ∪S)).
Using Mayer-Vietoris as above with Sreplaced by S, we find that
i∈I¯
Bc
ni[γi]=0inH1(R3\S).
The inductive step implies that the niall vanish. This completes the proof in the
separating case.
Nonseparating case. We turn to case where αdoes not separate the component of
Sthat contains it. We continue to denote that component of Sby Sα. Observe that28
genus(Sα\A) +1=genus(Sα),
so genus(S)=g−1. We apply the inductive step to S(which has b−1<bbound-
ary circles) to find a linearly independent set {[γ
1],...,[γ
2g−2]} ⊂ H1(R3\S)sat-
isfying the conditions of the lemma with Sin place of S. For every γ
ithat is not
contained in Bor in ¯
Bc, there exists a component U
i⊂∂B \Swith the signed inter-
section properties postulated by the inductive step.
As in the previous case, we can assume that the cycles are disjoint from Uε(D),
and thus lie in W. So, we may take
γ1:=γ1, ..., γ
2g−2:=γ
2g−2,
and, as before, set Ui:= U
i\Dor U
idepending on whether D⊂U
ior not (respec-
tively). We further define γ2g−1⊂¯
Bcto be αshifted slightly into the non-compact
component of R3\(Sα∪¯
B). Finally, we define γ2gto be a loop in the compact com-
ponent enclosed by Sαwith the property that γ2gintersects the disk Dtransversely
and in precisely one point (it is easy to find such a curve thanks to the non-separating
hypothesis); we take U2g:=D◦.
We claim that the loops γ1,...,γ
2gsatisfy the assertions of the lemma. The two
bullet points are easily checked by the construction of γ2g−1,γ2gand the assumption
28For a connected compact surface Mwith ∂M consisting of two circles, and the surface Mformed by
gluing these two boundary circles together, the inclusion-exclusion principle implies χ(M) =χ(M).
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196 O. Chodosh et al.
that the curves obtained via the inductive step avoid Uε(D). The other two claims in
the assertion follow by essentially the same argument as in the separating case.
This completes the proof.
Lemma 11.8 Suppose that S⊂R3is a closed and embedded genus-gsurface which
is transverse to a sphere ∂B ⊂R3.Denote W:=R3\S.
Assume that we are given {[γ1],...,[γ2g]} ⊂ H1(W ) ≈Z2gwhich is linearly in-
dependent and where each γisatisfies one of the following conditions:
•γiis contained in Bor in ¯
Bc,or
•there is a component Uiof ∂B \Sthat has non-zero signed intersection with γi
and zero signed intersection with each previous γj,j<i.
Then,at least one of the γiis contained in B,provided genus(S ∩B) > 0.29
Proof Note that, since genus(S ∩B) > 0, Lemma 11.7 implies (among other things)
that there is η⊂B\Sso that [η]=0inH1(W ) and so that for any m∈Z\{0},mη
is not homologous in ¯
B\Stoacyclein∂B \S.
Now, assume that none of the γidescribed above are contained in B. We claim
that
{[γ1],...,[γ2g],[η]}⊂ H1(W ) ≈Z2g
is a linearly independent set. This is impossible, so we will have proven the lemma.
To this end, assume that there are coefficients so that
m[η]=
2g
i=1
ni[γi]in H1(W ).
As in Lemma 11.7, by working downwards from i=2gand considering the intersec-
tion of each γiwith appropriate components of W∩∂B,usingtheUi’s, we can show
that ni=0 unless γiis contained entirely in ¯
Bc. As in the proof of Lemma 11.7,
applying Mayer–Vietoris to the pair (W ∩¯
B,W ∩Bc), we find that mη must be
homologous in ¯
B\Stoacyclein∂B \S. This contradicts the above choice of ηun-
less m=0, but in this case this contradicts the linear independence of the [γi].This
completes the proof.
Appendix A: Geometry of asymptotically conical shrinkers
Consider a shrinker n⊂Rn+1that is asymptotic to a smooth cone C.In[31,
Lemma 2.3], it was shown that the function w:C\BR(0)→Rparametrizing the
end of , i.e., such that
graphCw:={x+w(x)νC(x):x∈C\BR(0)}⊂,
29We do not need this here, but with minor modifications one can show that at least 2genus(S ∩B) curves
γiare contained entirely in B.
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Mean curvature flow with generic initial data 197
must satisfy w=O(r−1),∇(k)
∂rw=O(r−1−k), and ∇(k)w=O(r−1−k+η)for any
η>0. Here, r=|x|is the radial coordinate on the cone. The sharp asymptotics of w
(which we need in this paper) are, in fact:
Lemma A.1 The function wabove satisfies ∇(k)
Cw=O(r−1−k)as r→∞.
Proof We prove this for k=1—higher derivatives follow by induction. The shrinker
equation (2.4) along implies (using our curvature conventions from Sect. 2.8) that
H(x+w(x)νC(x)) +1
2x+w(x)νC(x), ν=0.
Moreover, by [31,(C.1)]wehave
ν(x)
=(1+|(Id −w(x)AC(x))−1∇w(x)|2)−1
2(−(Id −w(x)AC(x))−1∇w(x)+νC(x)).
By combining these equations we find
r∇∂rw(x)−w(x)=W(x),
where
W(x):= 2(1+|(Id −w(x)AC(x))−1∇w(x)|2)1
2H(x+w(x)νC(x)).
We have used the fact that AC(∂r,·)≡0, as well as that Id−wACis an endomorphism
of TCand νC⊥TC. Observe that
∇(k)W=O(r−1−k). (A.1)
Indeed, ∇(k)H=O(r−1−k), while the other terms decay at a faster rate. For x=rp
for p∈, the link of C, choose a vector ϑ∈Tp. Extend ϑto be parallel along
γ:r→rp. Note that [rϑ,∂r]=0, so by (A.1) we find:
r∇∂r(∇rϑw) −∇rϑw=∇rϑW=O(r−1).
Integrating this from infinity (cf. [31, Lemma 2.3]), we find ∇rϑw=O(r−1).As
ϑ=O(1), we find that ∇w=O(r−2)(decay of the radial component was shown in
[31, Lemma 2.3]).
Using this improved decay, one can set η:= 0in[31, Corollary 2.4], [31,
Lemma 2.5], [31, Lemma 2.7], [31, Lemma 2.8]. Thus, we have:
Lemma A.2 The second fundamental form of satisfies,for k≥0,
|∇(k )
C(A◦F−AC)|=O(r−3−k)
as r→∞.Here,F:x→x+w(x)νC(x)parametrizes the end of over C.
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198 O. Chodosh et al.
Corollary A.3 The second fundamental form of satisfies,for k≥0,
|∇(k )
A(xT,·)|=O(r−2−k)
as r→∞.Here,xTis the projection of the ambient position vector x∈to Tx.
Appendix B: Non-standard Schauder estimates
We recall the following non-standard Schauder estimate due to Knerr:
Theorem B.1 ([84,Theorem1]) Suppose that B2⊂Rnand we are given coefficients
aij ,bi,c:B2×[−2,0]→Rand functions u,h:B2×[−2,0]→Rso that uis a
classical solution of
∂
∂t u−aij ∂2
∂xi∂xju−bi∂
∂xiu−cu =h.
Assume
sup
t∈[−2,0]aij (·,t)0,α;B2+bi(·,t)0,α;B2+c(·,t)0,α;B2≤
and
aij (x, t )ξiξj≥λ|ξ|2,∀ξ∈Rn,
with λ,∈(0,∞).Then,for T∈[−1,0],
2
j=0Dj
xu0,α,α/2;B1×[−1,T ]+sup
t∈[−1,T ]∂
∂t u(·,t)0,α;B1
≤Csup
t∈[−4,T ]u(·,t)0;B2+h(·,t)0,α;B2,(B.1)
where C=C(n,α,λ,).Here,·0,α,α/2denotes the standard parabolic spacetime
Hölder norm and Dj
xudenotes the matrix of jpartial derivatives in spatial directions.
Note that this differs from the standard Schauder estimates because we’re only
assuming Hölder continuity on aij ,bi,c,hin the space directions. As a result, we
only get a spatial Hölder bound on ∂
∂t u. The other Hölder bound remain as in the
standard Schauder theory.
We also have the following variant:
Corollary B.2 Assume the setup of Theorem B.1.Then,for T∈[−1,0],
2
j=0Dj
xu0,α,α/2;B1×[−1,T ]+sup
t∈[−1,T ]∂
∂t u(·,t)0,α;B1
≤CuL1(B2×[−2,T ])+sup
t∈[−4,T ]h(·,t)0,α;B2,(B.2)
where C=C(n,α,λ,).
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Mean curvature flow with generic initial data 199
Proof For simplicity, let’s prove this for T=0. Let us consider the seminorm
[u]∗
Br×[−r2,0]:=[D2
xu]α,α/2;Br×[−r2,0].
By interpolation and integrating along line segments, we can show that for each ε>0
there exists C=C(n, α,ε) such that
sup
t∈[−4,0]u0;B2≤ε[u]∗
B2×[−4,0]+CuL1(B2×[−4,0]).
Thus, (B.1) implies
[u]∗
B1×[−1,0]≤ε[u]∗
B2×[−4,0]+CuL1(B2×[−4,0])+sup
t∈[−4,T ]h(·,t)0,α;B2,
(B.3)
where C=C(n, α,ε,λ,). By scaling down to parabolic balls Br×[−r2,0]and
also recentering in space and time, we obtain
r2+α[u]∗
Br(y0)×[t0−r2,t0]≤εr2+α[u]∗
B2r(y0)×[t0−4r2,t0]+γ,
where
γ:=CuL1(B2×[−4,0])+sup
t∈[−4,T ]h(·,t)0,α;B2
is just the second term of the right hand side of (B.3). We now apply the absorption
lemma due to L. Simon, [94, Lemma, p. 398] on the monotone subadditive function
S(Br(y0)×[t0−r2,t
0]):= [u]∗
Br(y0)×[t2
0−r2,t0], with scaling exponent 2 +α.(Note
that this monotone subadditive function extends trivially to convex sets.) By L. Si-
mon’s absorption lemma, we can choose εsmall enough depending on n,α, such
that
[u]∗
B1×[−1,0]≤CuL1(B2×[−4,0])+sup
t∈[−4,0]h(·,t)0,α;B2,
where C=C(n,α,λ,). This yields (B.2): the first summand of the left hand side
is obtained by interpolation, and the second by reusing the parabolic PDE.
Appendix C: Brakke flow uniqueness of regular mean curvature flows
We include here the following uniqueness result for Brakke flows.
Proposition C.1 Suppose that Mis an integral unit-regular Brakke flow in Rn+1×
[t0,t
2]and [t0,t
2]t→M(t) is a smooth mean curvature flow with
lim
r→0sup
x∈M(t)
M(t)((x,t),r)=1
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200 O. Chodosh et al.
for all t∈(t0,t
2]and
sup
(x,t)∈Rn+1×[t0,t2]|AM(t)(x)|<∞.
If M(t1)=Hn$M(t1)for some t1∈(t0,t
2]then M(t) =Hn$M(t) for all t∈[t1,t
2].
Proof The monotonicity formula and unit regularity property (cf. [108]) of Mim-
plies that Mis the multiplicity one Brakke flow associated to a smooth flow for some
interval t∈[t1,t
1+η].Asin[8, Proposition 4.4] (following [53,55]), this flow is a
smooth graph over M(t) and has bounded curvature; thus M(t) =Hn$M(t) by e.g.
[30]or[54].
We can thus conclude via a continuity argument. Let T∈(t1,t
2]denote the first
time the assertion fails. Suppose that T<t
2. Unit regularity and the assumptions
about M(t) imply that M(T ) =Hn$M(T). Thus, we can repeat the previous argu-
ment to conclude that M(t) =Hn$M(t) for t∈[T,T +η]for some η>0. This is
a contradiction, completing the proof.
Appendix D: Ilmanen’s localized avoidance principle
In this section we will give a proof of Ilmanen’s localized avoidance principle for
mean curvature flow. The proof is a parabolic version of the barrier principle and
moving around barriers in [77].
Let be an open subset of Rn+1×R, and let ⊂Rn+1×Rbe relatively closed
in . We call abarrier (resp. strict barrier) for mean curvature flow in provided
that, for every smooth open set E⊂\and for every (x,t) ∈∂E ∩∩with
∇∂Et(x,t)= 0, we have
f(x,t)≤H∂E(t) ·ν(x,t) (D.1)
(resp. f(x,t)< H∂E(t) ·ν(x,t)), where H∂E(t) the mean curvature vector of ∂E(t),
ν(x,t) is the inward normal of ∂E(t) at x, and fνis the normal speed of the evolution
t→∂E(t) in a neighborhood of (x,t).
Let W⊂Rn+1×Rbe open and let u:W→Rbe smooth, positive, bounded and
such that uvanishes on ∂W(t) for all t∈t(W ).Forp,q∈W(t), define the distance
dt(p,q):=inf γ
u(γ (s ), t)−1ds :γis a curve joining p,qin W(t)
.(D.2)
We assume that, for each t∈t(W ), the distance dtis complete. We use the standard
convention that inf ∅=∞. Note that dtis just the distance in the (complete) con-
formally Euclidean metric gt:=u(·,t)
−2gRn+1. More generally, we can consider the
distance between two closed sets in Wtdefined in the usual way. For U⊂W,Uopen,
define
Ur=(x,t)∈U:dt(x,∂U(t)) > r .
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Mean curvature flow with generic initial data 201
Define the degenerate second order elliptic operator
Ku(x,t)=inf
StrSD2u(x,t),
where Sranges over all n-dimensional subspaces of Rn+1.
Lemma D.1 Suppose that W\Uis a barrier in Wand u:W→Ris as above,with
ut−Ku ≤0(resp.<0).
Then W\Uris a barrier (resp.a strict barrier)in W.
Proof Let EWbe a smooth open set with
E⊂Urand (x,t)∈∂E ∩(W \Ur).
We have to show that (D.1) holds. Define
Es:={(x,t)∈W:dt(x,¯
E) < s},F:=Er.
Then ¯
Fis compact, F⊂U, and ∂F meets ∂U.
We fix γ(t) to be a shortest gt-geodesic from ∂E(t) to ∂U(t) with endpoints
x∈E(t) and y∈∂F(t) ∩∂U(t). Thus the normal exponential map of ∂E(t) with
respect to gthas no focal points along γ(t) \{y}. Note that this also holds for the
normal exponential map of ∂E(τ) with respect to gτin a spacetime neighborhood
of γ\{y}. Therefore in a spacetime neighborhood of γ\{y},(τ , s) → ∂Es(τ ) is
smooth and smoothly varying. For τclose to t,letx(τ ) be the normal evolution of x
along τ→∂E(τ) such that x(t ) =x.Forτclose to twe define γ(τ)to be the normal
g(τ)-geodesic starting at x(τ), i.e. the normal evolution of x(τ) along s→∂Es(τ ).
Note that
1=gτ(γ(τ ), γ(τ )) =u−2gRn+1(γ(τ ), γ(τ )) =⇒ |γ(τ )|gRn+1=u.
We denote x(τ, s ) =γ(τ , s) and fτν∂Es(τ ) to be the normal velocity of the evolution
τ→∂Es(τ ) in Rn+1. Furthermore, note that the gτ-length of γ(τ , ·)satisfies
gτγ(τ, [0,s])=s
and thus
0=d
dτ gτγ(τ, [0,s])=d
dτ γ(τ,[0,s ])
dgτ=d
dτ γ(τ,[0,s ])
u−1dgRn+1
=−u(x(τ, s ))−1f(x(τ , s), τ ) +u(x(τ , 0))−1f(x(τ, 0), τ )
−γ(τ,[0,s ])
u−2uτdgRn+1
=−u(x(τ, s ))−1f(x(τ , s), τ ) +u(x(τ , 0))−1f(x(τ, 0), τ ) −s
0u−1uτdgτ.
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202 O. Chodosh et al.
Differentiating the last equation in syields
∂
∂s f=−uτ−fDu·ν∂Es(t) .(D.3)
Similarly, looking at the evolution of s→∂Es(t) in Rn+1we have
∂
∂s H∂Es(t) =−∂Es(t) u−|A∂Es(t) |2u
=−trSD2u−H∂Es(t) Du ·ν∂Es
t−|A|2u
≤−Ku−H∂Es(t) Du ·ν∂Es(t) .
(D.4)
Combining (D.3), (D.4) we see that ψ:=f−Hsatisfies, along γ(t ),
∂
∂s ψ≥−Cψ . (D.5)
We first assume that y(t) is not a focal point of the exponential map of ∂E(t).This
implies that Fis locally smooth around yand ∇∂F t(y,t)= 0. If ψ(0)>0 then (D.5)
implies that ψ(r) > 0 which gives a contradiction to the assumption that W\Uis a
barrier. If ψ(0)≥0 and ut−Ku < 0 then likewise ψ(r) > 0 which again yields a
contradiction, proving that P\Uris a strict barrier.
If the normal exponential map of ∂E(t) focuses at y(t ), then we may approximate
Eby E⊂Esuch that E∩∂Ur={x},yis not a focal point and such that in the
above argument we can replace Eby E.
Lemma D.2 If is an open subset of spacetime and Mis a closed weak set flow in
,then Mis a barrier in .
Proof Assume E⊂\Mis open and smooth, and at (x0,t
0)∈∂E ∩M∩we
have
∇∂Et(x0,t
0)=0,f(x0,t
0)>H
∂E(t)(x0,t
0), (D.6)
where H∂E(t)(x,t)=H∂E(t) ·ν∂E(t) and ν∂E(t) is the inward pointing unit normal of
∂E(t). We can furthermore assume that ∂E ∩M={(x0,t
0)}. For small r>0, (D.6)
implies that
f>H
∂E(t) on Br(x0)×[t0−r2,t
0],(D.7)
and that ∂E(t) is C2-close to an n-dimensional plane for all t∈[t0−r2,t
0].We
can thus solve mean curvature flow S=(S(t))t∈[t0−r2,t0]with the induced parabolic
boundary conditions on ∂E∩Br(x0)×[t0−r2,t
0]. Note that t→∂E(t)∩Br(x0)is a
barrier for Sfrom one side, in view of (D.7). Thus Shas to run into M, contradicting
that Mis a weak set flow. Thus, (D.6) fails, and the result follows.
We can now state and prove Ilmanen’s localized avoidance principle. For R,α≥0,
and (x0,t
0)∈Rn+1×R,weset
uα(x,t):=(R2−|x−x0|2−(2n+α)(t −t0))+(D.8)
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Mean curvature flow with generic initial data 203
on R×Rn+1. Note that for α>0:
∂
∂t uα(x,t)<Ku
α(x,t), (D.9)
for all (x,t) with uα(x,t)> 0.
Theorem D.3 (Ilmanen) Consider two closed weak set flows M,Min Rn+1and
constants satisfying R>0, γ>0, a<b<a+R2−γ
2n.Assume that
M(t) ∩Bγ+R2−2n(t−a)(x0)and M(t) ∩Bγ+R2−2n(t −a)(x0)
are disjoint for t∈[a, b).Then,using this choice of Rand x0along with t0=aand
α=0in (D.8), we have that t→ dt(M(t), M(t )) is non-decreasing for t∈[a,b)
and
M(b) ∩M(b) ∩BR2−γ−2n(b−a)(x0)=∅.
Before proving Theorem D.3, let us indicate how we plan to apply it. If M(a),
M(a) are disjoint and one knows apriorithat
M(t) ∩M(t) ∩(BR2+γ−2n(t −a)(x0)\BR2−γ−2n(t −a)(x0)) =∅,
for t∈[a, b], then Theorem D.3 and a straightforward continuity argument imply that
M(t) ∩M(t) ∩BR2−γ−2n(t −a)(x0)=∅
for t∈[a,b]. In other words, if the two weak set flows are disjoint near the boundary
of the comparison region, then they remain disjoint.
Proof of Theorem D.3 We first note that the assumptions imply that for sufficiently
small α>0, the distance dα
twith respect to (uα(·,t))
−2gRn+1between Mand M
is attained away from the boundary of the set W:= {uα(x,t)> 0}for all t∈[a,b).
Assume that dα
a(M,M)=r>0. We can thus argue as in [73,C1,1Interposition
Lemma] find a C1,1hypersurface in ({uα(x,a) >0},g
a)separating Mand M,
such that dα
a(M,) =dα
a(M,) =r/2, with both distances attained away from
the boundary of W. Consider =∩BR−η(x0)for suitable small η>0 such
that has smooth boundary. Solve smooth mean curvature flow
tstarting at
with fixed Dirichlet boundary conditions for a≤t<a+εfor small ε>0. We can
assume that dα
t(M,
t)and dα
t(M,
t)are attained away from the boundary ∂
for all a≤t<a+ε. Choose U=W\M. Note that for 0 <s<r/2wehave
∂Us(a) ∩=∅. So by Lemma D.1 and Lemma D.2, together with (D.9)wehave
that (t) ⊂Us(t) for all 0 <s<r/2 and a≤t<a+ε. This implies that the dis-
tance dα
tbetween Mand
tis non-decreasing on [a,a +ε). We can argue simi-
larly to see that the distance dα
tbetween Mand
tis non-decreasing on [a,a +ε).
Thus dα
t(M,M)is non-decreasing for t∈[a,a +ε). But the monotonicity formula
implies that dα
a+ε(M,M)≥lim supt→(a+ε)+dα
t(M,M). Thus a direct continuity
argument implies that dα
t(M,M)is non-decreasing for t∈[a, b). Letting α→0
gives the result.
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204 O. Chodosh et al.
We note that this implies a well-known Frankel property for self-shrinkers. For
completeness, we state our result in full generality, in the context of F-stationary
varifolds, i.e., varifolds in Rn+1that are stationary for the conformally Euclidean
metric in Sect. 2.7 whose stationary points coincide with self-shrinkers.
Corollary D.4 (Frankel property for shrinkers) If V,Vare F-stationary varifolds,
then suppV∩supp V=∅.
Proof If supp V∩supp V=∅, then the associated self-similarly shrinking Brakke
flows M,Msatisfy
suppM(t ) ∩suppM(t) =∅,t<0.
Applying Theorem D.3 with a=−1, b=0, R>√2n, and recalling that the support
of the spacetime track of a Brakke flow is a weak set flow [74, 10.5] we arrive at a
contradiction; indeed, 0∈supp M(0)∩supp M(0).
Appendix E: The Ecker–Huisken maximum principle
For the reader’s convenience, we recall here a special case of the variant of the Ecker–
Huisken maximum principle (see [54]) proven in [8], which we’re going to make use
of:
Theorem E.1 ([8,TheoremA.1]) Suppose that {t}t∈[a,b) is a smooth mean curvature
flow in Rn+1\BR,with ∂t⊂∂BR.Assume that uis a C2function on tso that
(1) it satisfies
∂
∂t −tu≥a·∇tu+bu
with supt∈[a,b) supt|a|+|b|<∞,
(2) u>0on the parabolic boundary a∪(∪t∈[a,b)∂t),
(3) for all t∈[a,b),and
t|u|2+|∂u
∂t |2+|∇tu|2+|∇2
tu|2ρ(0,b)dHn<∞,
where
ρ(0,b)(x,t)=(4π(b −t))−n
2e−|x|2
4(b−t) .
Then,for all t∈[a,b),inf
tu≥0.
Appendix F: Weak set flows of cones
For this appendix, the reader might find it useful to recall the notions set forth in
Sect. 2. We collect results of [89] on weak set flows and outermost flows and show
that they are also applicable (with minor modifications) to the flow of hypercones.
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Mean curvature flow with generic initial data 205
Proposition F.1 ([89, Proposition A.3]) Suppose that Fis any closed subset of Rn+1,
and let M⊂Rn+1×R+be its level set flow.Set:
M(t) :={x∈Rn+1:(x,t)∈∂M}.
Then t→M(t) is a weak set flow.
In what follows, we consider Fto be the closure of its interior in Rn+1and satisfy
∂F =∂Fc.
We call such a set Fadmissible.30 Let F:=Fc, denote the level set flows of F,F
by M,M, and set F(t):= M(t),F(t ) := M(t). In line with Proposition F.1,we
set:
M(t) :={(x,t)⊂Rn+1:x∈∂M},
M(t) :={(x,t)⊂Rn+1:x∈∂M}.
(Here ∂M,∂Mare the relative boundaries of M,Mas subsets of Rn+1×R+).
We call
t→M(t), t → M(t)
the outer and inner flows of M:=∂F. By Proposition F.1,M(t),M(t ) are contained
in the level set flow generated by M. Furthermore,
M(t) =lim
τt∂F(τ)
for all t>0, and M(t) =∂F(t) for all but countably many t. See [89, Theorems
B.2, C.1]. Note that [89, Theorems B.2] directly carries over to M=∂F where Fis
admissible.
Let ⊂Sndenote a fixed smooth, embedded, closed hypersurface. Consider the
equidistant deformations (s)−ε<s<ε of ⊂Snfor some consistent choice of normal
orientation. We further consider the regular hypercone C=C() and the smooth
perturbations Cs=C(s). Note that Csdivides Rn+1into two open sets ±
ssuch
that Cs=∂±
sas well as C() ∩+
s=∅for s>0 and C() ∩−
s=∅for s<0.
We now consider,
s,r :=∂+
s\Br(0),
for 0 <r <1 and s>0. We denote with ˜
s,r a smoothing of s,r that rounds off the
corners near ∂Br(0). Similarly we set:
s,r :=∂(+
s∩B1/r(0)) \Br(0)
30Note that this slightly extends the definition in [89], where ∂F (∂U in their notation) would be a compact,
smooth hypersurface. This extension allows us to flow from non-compact and non-smooth initial surfaces.
This does not change anything in the analysis of [89].
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206 O. Chodosh et al.
for 0 <r <1, s>0, and ˜
s,r to be a smoothing of
s,r that rounds off its corners.
Note that by using the smoothings ˜
s,r we can construct compact regions Fi⊂+
with smooth boundaries such that
(1) For each i,Fiis contained in the interior of Fi+1.
(2) ∪Fi=+.
(3) Hn$∂Fi→Hn$C().
By perturbing Fislightly, we can also assume that
(4) the level set flow of ∂Finever fattens.
We then directly generalize [89, Theorems B.3, B.5]. The proof extends verbatim.
Theorem F.2 There is an integral unit-regular Brakke flow t∈[0,∞)→ μ(t ) such
that μ(0)=Hn$C() and such that the spacetime support of the flow is the space-
time set swept out by t∈[0,∞)→M(t),where t→ M(t) is the outer flow of C().
That is,for t>0, the Gaussian density of the flow μ(·)at (x,t) is >0if and only if
x∈M(t).
Remark (i) Note that by uniqueness of the level set flow, the outer flow satisfies
M(t) =√tM(1). Together with unit regularity and White’s local regularity theory
this implies that there is a R0=R0() such that the Brakke flow constructed in The-
orem F.2 is a smooth expanding solution, agreeing with M(t), outside of B√tR0(0)
for all t>0.
(ii) Theorem F.2, and all of the above, applies also to the inner flow.
Appendix G: Brakke flows with small singular set
In this section we show that if a Brakke flow has small singular set, then the regular set
is connected, provided it is connected in a neighborhood of the initial time. To prove
this, we show that for a closed set S⊂Rn+k×R, a Brakke flow (with bounded area
ratios) on (Rn+k×R)\Sextends across Sprovided Shas vanishing n-dimensional
parabolic Hausdorff measure.31
Remark In [39, Claim 8.4] it was observed that the classification of low entropy
ancient flows implies connectivity of the regular part of a flow in R3with only
(multiplicity-one) spherical and cylindrical singularities, by an argument similar to
Kleiner–Lott’s proof [83, Theorem 7.1] that a singular Ricci flow of 3-manifolds has
only finitely many bad world lines. We show here that one can prove connectivity
under considerably weaker hypothesis. We note that our approach has no hope of es-
timating the number of bad world lines. It would be interesting to study the Hausdorff
dimension of bad world lines in a k-convex mean curvature flow in Rn+1.
31See also [33, Appendix D] where it is shown that an integral 2-dimensional Brakke flow in R3\{0}with
bounded area ratios extends across the origin.
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Mean curvature flow with generic initial data 207
We first recall a well known extension theorem for varifolds, originally considered
by de Giorgi–Stampacchia [51].
Lemma G.1 Let Vbe a rectifiable n-varifold in Rn+kwith bounded area ratios,i.e.,
V(Br(x)) ≤Crn.If S⊂Rn+kis closed,Hn−1(S) =0, and the restricted varifold
V:=V$(Rn+k\S) has absolutely continuous first variation H∈L1
loc(Rn+k;μV),
then Vhas absolutely continuous first variation equal to H,too.
Proof Without loss of generality, we assume that Sis compact. For δ>0, we can
find balls {Bri(xi)}N
i=1covering Swith
N
i=1
rn−1
i<δ.
Choose cut-off functions 0 ≤ξi≤1 with ξi≡1 outside of B2ri(xi),ξi≡0onBri(xi),
and |Dξi|≤2
ri. Then, set ξδ=N
i=1ξiand note that
|Dξδ|≤
N
i=1
2
ri
χBri(xi).
For a vector field #∈C1
c(Rn+1),wehave
ξδdivM#dμ
V+ξδ#·HdμV=−DTξδ·#dμ
V.
Note that
DTξδ·#dμ
V≤
N
i=1
2
riV(Bri(xi))#L∞≤Cδ.
Sending δ→0, the dominated convergence theorem implies
divM#dμ
V=−#·HdμV.
Thus, δV is absolutely continuous with respect to dμVand, since μV(S) =0, the
generalized mean curvature of Valso equals H. This completes the proof.
We now extend this to Brakke flows (recall our conventions in Sect. 2.4).
Theorem G.2 Consider (μ(t ))t∈Ibe a 1-parameter family of Radon measures on
Rn+kand S⊂Rn+k×Ra closed set with Hn
P(S) =0. Assume that
(1) The measures μ(t) have uniformly bounded area ratios,i.e., μ(t)(Br(x)) ≤Crn.
(2) For almost every t∈I,there exists an integral n-dimensional varifold V(t)
with μ(t) =μV(t) so that V(t ) =V(t)$(Rn+k\S(t)) has absolutely continu-
ous first variation in L1
loc(Rn+k;dμV(t) )and has mean curvature Horthogonal
to Tan(V (t ), ·)almost everywhere.
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208 O. Chodosh et al.
(3) For any compact set K⊂(Rn+k×R)\S,we have
K
(1+|H|2)dμ(t )dt < ∞.
(4) If [t1,t
2]⊂Iand f∈C1
c((Rn+k×[t1,t
2])\S) has f≥0, then
f(·,t
2)dμ(t
2)−f(·,t
1)dμ(t
1)
≤t2
t1−|H|2f+H·∇f+∂
∂t fdμ(t)dt.
Then (μ(t))t∈Iis a Brakke flow on Rn+k.
Proof It suffices to prove this for Scompact. We begin by defining the relevant cutoff
function. Choose a family of parabolic balls
Pri(xi,t
i)=Bri(xi)×(ti−r2
i,t
i+r2
i),
where i=1,...,N, so that S⊂∪N
i=1Pri(xi,t
i)and
N
i=1
rn
i<δ.
For each parabolic ball, choose a cutoff function 0 ≤ζi≤1 so that ζi≡1on
P2ri(xi,t
i)and ζi≡0onPri(xi,t
i). We can assume that |Dζi|≤C/riand |∂
∂t ζi|≤
C/r2
i. Set
ζδ=min
iζi
and define a mollified function ζδ,ε as follows. Choose 0 ≤ϕ1,ϕ
n+k≤1 standard
mollifiers on R,Rn+kand set
ζδ,ε(x,t)=Rn+k×Rε−n−k−2ϕn+k(ε−1(x−y))ϕ1(ε−2(t −s))ζδ(y,s)dyds.
We now estimate the derivatives of ζδ,ε.
Claim
lim sup
ε→0|∂
∂t ζδ,ε(x,t)|≤C
N
i=1
1
r2
i
χP2ri(xi,ti).(G.1)
Proof of (G.1)Wehave
∂
∂t ζδ,ε(x,t)
=Rn+k×Rε−n−k−4ϕn+k(ε−1(x−y))ϕ
1(ε−2(t −s))ζδ(y,s)dyds
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Mean curvature flow with generic initial data 209
=Rn+k×Rε−n−k−4ϕn+k(ε−1(x−y))ϕ
1(ε−2(t −s))(ζδ(y,s)−ζδ(y,t))dyds
≤Cmax
isup
(y,s)∈Pε(x,t ) |∂
∂t ζi(y,s)|
≤C
N
i=1
sup
(y,s)∈Pε(x,t ) |∂
∂t ζi(y,s)|,
which implies the inequality follows after sending ε→0.
Claim
lim sup
ε→0|Dζδ,ε(x,t)|2≤C
N
i=1
1
r2
i
χP2ri(xi,ti).(G.2)
Proof of (G.2) As in the proof of (G.1), we find
Dζδ,ε(x,t)
2≤Cmax
isup
(y,s)∈Pε(x,t ) |Dζi(y,s)|2≤C
N
i=1
sup
(y,s)∈Pε(x,t ) |Dζi(y,s)|2.
This implies the claim, as before.
Now, for 0 ≤f∈C2
c(Rn+k×[t1,t
2])we consider ζ2
δ,εfin (4) above. We find
ζδ,ε(·,t
2)2f(·,t
2)dμ(t2)−ζδ,ε(·,t
1)2f(·,t
1)dμ(t1)
≤t2
t1−|H|2ζ2
δ,εf+ζ2
ε,δH·∇f+ζ2
ε,δ ∂
∂t fdμ(t)dt
+t2
t12ζε,δfH·∇ζε,δ +f∂
∂t ζ2
ε,δdμ(t)dt
≤t2
t1−(1−γ)|H|2ζ2
δ,εf+ζ2
ε,δH·∇f+ζ2
ε,δ ∂
∂t fdμ(t)dt
+t2
t1γ−1|∇ζε,δ |2+f∂
∂t ζ2
ε,δdμ(t)dt
≤t2
t1−(1−γ)|H|2ζ2
δ,εf+ζ2
ε,δH·∇f+ζ2
ε,δ ∂
∂t fdμ(t)dt
+Cγ−1fC1t2
t1
N
i=1
1
r2
i
μ(t)(Bri(xi))χ(ti−r2
i,ti+r2
i)(t)dt
≤t2
t1−(1−γ)|H|2ζ2
δ,εf+ζ2
ε,δH·∇f+ζ2
ε,δ ∂
∂t fdμ(t)dt
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210 O. Chodosh et al.
+Cγ−1δfC1
≤t2
t1−(1−2γ)|H|2ζ2
δ,εf+ζ2
ε,δ ∂
∂t fdμ(t)dt
+Cγ−1δfC1+Cγ −1D2fL∞.
In the final inequality, we have used [74, Lemma 6.6].
Sending δ→0, we can use Lemma G.1 (and Lemma G.3 below) to conclude
that for almost every t, the varifold V(t)has absolutely continuous first variation in
L1
loc(Rn+k,dμ(t)) and that
t2
t1K
(1+|H|2)dμ(t )dt < ∞
for any compact set Kand [t1,t
2]⊂I. Then, dominated convergence and the above
inequality guarantees
f(·,t
2)dμ(t2)−f(·,t
1)dμ(t1)
≤t2
t1−(1−γ)|H|2f+H·∇f+∂
∂t fdμ(t)dt,
which implies (4) after sending γ→0. This completes the proof, after observing that
0≤f∈C1
ccan be approximated by 0 ≤f∈C2
c.
Lemma G.3 Suppose that S⊂Rn+k×Ris a closed set with Hn
P(S) =0. Then for
almost every t,
Hn−2(S(t )) =0,
where S(t ) =S∩t−1(t).
Proof As usual, we can assume that Sis compact. Choose parabolic balls Pri(xi,t
i)
covering Swith ri<δ and !irn
i<δ. Set I(t) := {i:t∈(ti−r2
i,t
i+r2
i)}and note
that
S(t) ⊂
i∈I(t)
Bri(xi).
Note that
t2
t1
i∈I(t)
rn−2
idt =t2
t1
i
rn−2
iχ(ti−r2
i,ti+r2
i)(t)dt =2
i
rn
i<2δ.
This proves that
|{t∈[t1,t
2]:Hn−2
δ(S(t )) > ε}|<Cδ
ε.
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Mean curvature flow with generic initial data 211
Because Hn−2
δ(S(t )) is non-decreasing as δ0, we thus see that
|{t∈[t1,t
2]:Hn−2(S(t )) > ε}|=0.
Sending ε→0 completes the proof.
For a Brakke flow M, define "regMto be the set of points (x,t) so that there is
ε>0 with
M$(Bε(x)×(t −ε2,t])=kHn$M(t),
where kis a positive integer and M(t) is a smooth mean curvature flow. Note that
points in reg Mare defined similarly, but with k=1; thus, regM⊂"regM.
Corollary G.4 Consider M=(μ(t))t∈Ia unit-regular integral n-dimensional Brakke
flow in Rn+kwith μ(t) =Hn$M(t) for t∈[0,δ),where M(t) is a mean curvature
flow of connected,properly embedded submanifolds of Rn+kand δ>0. If
Hn
P(suppM\"regM)=0,
then "reg M=reg Mand reg Mis connected.
Proof We claim that M(0)= ∅ for any a connected component, M,of "regM.
From this, we immediately have that the multiplicity on this component is k=1,
so "reg M=reg M. Moreover, since M(t) is connected for t∈[0,δ), we also will
have regMis connected.
Now, consider Mas above. Set
M:=M∪(supp M\"reg M). Theorem G.2 im-
plies that μ(t)$
M(t) is a Brakke flow. However, if M(0)=∅, then we can apply
Huisken’s monotonicity formula to conclude that μ(t)$
M(t) =0 for all t.Thisisa
contradiction, completing the proof.
Combining White’s parabolic stratification [104, Theorem 9] with the previous
corollary this implies:
Corollary G.5 Suppose that Mis a unit-regular integral n-dimensional Brakke flow
in Rn+kwith μ(t) =Hn$M(t) for t∈[0,δ),where M(t) is a mean curvature flow of
connected,properly embedded submanifolds of Rn+kand δ>0. Assume that Mhas
the following properties:
(1) If there is a static or quasi-static planar tangent flow at X,then X∈"reg M.
(2) There are no static or quasi-static tangent-flows supported on a union of half-
planes or polyhedral cones.
Then "reg M=reg Mis connected.
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212 O. Chodosh et al.
Appendix H: Localized topological monotonicity
In this appendix we localize some of the results from [103]. We say a closed subset
Mof a spacetime Rn+1×Ris a simple flow in an open set U⊂Rn+1with smooth
boundary and over a time interval I⊂R,orasimpleflowinU×Ifor short, if there
is a compact n-manifold M, with or without boundary, and a continuous map
f:M×I→Rn+1
so that:
(1) M(t) ∩U=f(M,t), where M(t ) :={x∈Rn+1:(x,t)∈M},
(2) fis smooth on M◦×I, where M◦:= M\∂M,
(3) f(·,t),t∈I, is an embedding of M◦into U,
(4) t→ f(M◦,t),t∈I, is a smooth mean curvature flow: (∂
∂t f(·,t))
⊥=H(·,t),
and
(5) f|∂M×Iis a smooth family of embeddings of ∂M into ∂U.
The following lemma is easily proven but we will use it repeatedly in the sequel.
Lemma H.1 If M⊂Rn+1×Ris a simple flow in U×[0,T]then we have a diffeo-
morphism
(U ×[0,T])\M≈(U \M(0)) ×[0,T]
that restricts to diffeomorphisms U\M(t) ≈U\M(0)along each fibre.
We recall some definitions from [103]. For M⊂Rn+1×[0,T],t∈[0,T],weset:
W[t]:=Mc∩t−1({t}), (H.1)
W[0,T]:=Mc∩t−1([0,T]). (H.2)
The results of [103] apply precisely to these W[t],W[0,T]. Since we wish to localize
some of these results to open subsets ⊂Rn+1with smooth boundary, we introduce
the following localized objects.
W[t]:=Mc∩∩t−1({t}), (H.3)
W[0,T]:=Mc∩∩t−1([0,T]). (H.4)
Note that, in this notation, W[t]=WRn+1[t]and W[0,T]=WRn+1[0,T].
The following is a localization of [103, Theorem 5.2].
Theorem H.2 Let Mbe a level set flow and ⊂Rn+1be an open set with smooth
boundary,so that Mis a simple flow in U×[0,T]for some tubular neighborhood
Uof ∂.Then:
(1) For every point Xin W[0,T],there is a time-like path in W[0,T]joining X
to a point Y=(y,0)at time 0.
(2) If X,Yare in different connected components of W[0],then they are in different
connected components of W[0,T].
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Mean curvature flow with generic initial data 213
Proof To prove (1), note that for X∈U×[0,T], it is not hard to construct such a
path (by the simplicity assumption). In general, by [103, Theorem 5.2(i)], we can find
a time-like path in Mcconnecting Xto time 0. If this path remains in ×[0,T ],
the claim follows. On the other hand, if the path does not remain in ×[0,T ], then
it must enter U×[0,T]at some point. In this case, we can stop and concatenate with
the path in U×[0,T]that exists by the fact that the flow is smooth in that region.
For (2), consider X, Y ∈W[0]that are in distinct connected components of
W[0], but in the same connected component of W[0,T]. First, consider the case
when at least one of the points, say Xis in a connected component Vof W[0]that
does not intersect the tubular neighborhood U. Because Mis simple in U×[0,T],
the component Vof Mc∩t−1([0,T])containing Vdoes not intersect U×[0,T];
thus, it is contained in ×[0,T]. As such, X,Yare in distinct components of
W[0]:=Mc∩t−1({0})but in the same component of W[0,T]:=Mc∩t−1([0,T ]).
This contradicts [103, Theorem 5.2(ii)]. Thus, both Xand Ymust be connected in
W[0]to U. As such we can assume below, without loss of generality, that X, Y ∈U.
Let us set up some notation. For each connected component Vof W[0], we write
VU:= V∩U(note that VUmay be disconnected). Write ∂VU=∂−VU∪∂+VU∪
∂MVU, where ∂−VU=(∂V ∩∂) \M(0),∂+VU=(V ∩∂U ∩) \M(0)and
∂MV=∂V ∩M(0)are distinct and ∂−VU(resp. ∂+VU) is relatively open in ∂
(resp. ∂U). Let V(X)=V(Y)denote the components of W[0]containing X,Y.
Because Xand Yare assumed to be in the same connected component of
W[0,T], they are in the same connected component of W[0]by [103, Theo-
rem 5.2(ii)]. Choose a path γ⊂W[0]between Xand Yso that γis transverse to
∂U ∪∂.For∗∈{X, Y }, we can assume that γdoes not intersect ∂+V(∗)U(we
might have to exchange the points ∗∈{X, Y }for some other point in V(∗)U). Indeed,
we can simply consider the last time that γintersects ∂+V(X)
Uand the earliest time
that γintersects ∂+V(Y)
Uand truncate γnear these times (to still have endpoints
in U).
Choose a curve η⊂W[0,T]from Yto Xso that η∩(U ×[0,T])⊂U×{0}
and consists of two arcs exiting Uthrough ∂+V(Y)
U∪∂+V(X)
U(with a single
transverse intersection with each). Concatenating γwith η, we can find a loop σ1in
W[0,T].By[103, Theorem 5.4], there is a homotopy of loops in W[0,T] between
σ1and a loop σ0in W[0]. Perturb σ0slightly so it is transverse to ∂U. By con-
struction and the simplicity of Min U×[0,T], the loop σ0has the property that
for ∗∈{X, Y }, the mod 2 intersection number of σ0with ∂+V(∗)Uis 1. This is a
contradiction.
The following is a localized version of [103, Theorem 5.4].
Theorem H.3 Let Mbe a level set flow and ⊂Rn+1be an open set with smooth
boundary,so that Mis a simple flow in U×[0,T]for some tubular neighborhood U
of ∂ in .Then,any loop in W[0,T]is homotopic to one in W[0].In particular
H1(W[0])→H1(W[0,T])
is surjective.
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214 O. Chodosh et al.
Proof Fix a cover :˜
W[0,T]→W[0,T]associated to ι0:W[0]→W[0,T].
Set ˆ
W:= Mc∩(U ×[0,T]). Note that ˆ
W⊂W[0,T]deformation retracts onto
WU[0]⊂ ˆ
W, by the assumption that Mis simple in U×[0,T]. Set W,k[0,T]:=
(W[0,T]∩Wk[0,T])∪ˆ
W, where Wk[0,T]=Wk∩t−1([0,T])(see Sect. 2.3 for
the definition of Wk). Because ˆ
Wdeformation retracts onto WU[0], we can find a lift
˜ι0:W,0[0,T]→ ˜
W[0,T].
In the remainder of the proof we inductively define lifts of ιk:W,k[0,T]→
W[0,T],
˜ιk:W,k[0,T]→ ˜
W[0,T],
so that ˜ιk|W,k−1[0,T ]=˜ιk−1. Having done so, we can fit these lifts together to produce
a lift ˜ι:W[0,T]→ ˜
W[0,T]; thus, the covering was trivial, completing the
proof.
Let Mbe a classical flow corresponding to F:M×[a, b]→Rn+1in Rn+1×
[0,T]disjoint from M(0)so that ∂M⊂Wk−1. Set M
:= M∩( ×[0,T]).
(Note that Mmight not intersect ∂ ×[0,T]transversely and there is no guarantee
that points in M
can be connected to a part of the heat boundary of M.)
Claim There is a unique lift φ:M
→˜
W[0,T]so that φ(X) =˜ιk−1(X) for all
points X∈M∩W,k−1[0,T].
Proof Fix X=F(p,t)∈M
. Choose an open set O×[0,T]so that ( \U)×
[0,T]⊂O,X∈O, and ∂Ois a small C∞perturbation of ∂ ×[0,T]intersecting
Mtransversely. Define
t0=inf{τ∈[a, t]:F(p ×(τ , t )) ⊂O}.
It is clear that F(p,t
0)∈W,k−1[0,T], so we can consider ˜γthe unique lift of the
curve γ:[t0,t]τ→F(p,τ)with ˜γ(t
0)=˜ιk−1(F (p , t0)). We then define φ(X) =
˜γ(t).
It is clear that φis continuous and does not depend on the choice of O. It remains
to check that φ(X)=˜ιk−1(X) for X∈M∩W,k−1[0,T]. Choose Oas above and
let Vdenote the connected component of M∩W,k−1[0,T]∩Ocontaining X.The
argument in [103, Lemma 5.3] can be easily adapted to show that Vcontains a point
Y∈∂M∪∂O⊂W,k−1[0,T]. Since φ(Y) =˜ιk−1(Y ), the maps agree on all of V.
This completes the proof of the claim.
Claim If M1,M2are two classical flows with heat boundaries in Wk−1and X∈
M1∩M2∩( ×[0,1])then φ1(X) =φ2(X).
Proof Given X, we can choose Oas above but with ∂Otransverse to M1and M2.
Now, as in [103, p. 328], the maximum principle guarantees that there is a connected
subset Kof M1∩M2containing Xand some point in ∂M1∪∂M2. Either K∩
∂O=∅, in which case there is Y∈(∂M1∪∂M2)∩K∩Oor K∩∂O= ∅,in
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Mean curvature flow with generic initial data 215
which case there is Y∈K∩∂O. Either way, Y∈W,k−1[0,T]. By the previous
claim, φ1(Y ) =˜ιk−1(Y ) =φ2(Y ). Because φ1|K,φ2|Kagree at Y, they must also
agree at X.
This completes the proof.
The following is a localized version of [103, Theorem 6].
Theorem H.4 Consider ⊂Rn+1an open set with smooth compact boundary and U
a fixed tubular neighborhood of ∂.Choose T0so that the mean curvature flow of ∂,
t→ ∂(t) remains smooth and inside some open set ˜
UUfor t∈[0,T
0].Then,
for any 0<T ≤T0,let Mbe a weak set flow in Rn+1that is simple in U×[0,T].
Then,
Hn−1(W[T])→Hn−1(W[0,T])
is injective.
Proof For 0 <T ≤T0fixed, suppose that [C]∈Hn−1(W[T])is a polyhedral (n −
1)-chain so that there is Pa polyhedral n-chain in W[0,T]with ∂P =C. We can
assume that the support of Pis disjoint from ˜
U∪{t=0}. Consider the projection
π(x,t) =(x,T). Set π#P=Pand note that ∂P=C. We aim to show that Pis
homologous (relative to its boundary) to a chain disjoint from M(T ).
Let Mbe the level set flow generated by . By the avoidance principle for weak
set flows (cf. [103, Theorem 4.1]), M(t) remains a positive distance from M(t) as
well as a positive distance from ∂(t). In particular, we can enlarge slightly to
to ensure that Mavoids some tubular neighborhood Uof ∂(so in particular, it is
asimpleflowinU×[0,T]).
Fatten M(T ) slightly to get a closed set Kin Rn+1×{T}that is disjoint from
˜
U∪M(T ) and has smooth boundary. If γis a loop in ( ×{T})\K, then by
Theorem H.3 applied to M,γis homologous in (×[0,T])\Mto a loop at
t=0. In particular, this means that the oriented intersection number of γwith P
(and thus P) is zero.
Now, assign each component of
(×{T})\(K ∪P)
a multiplicity so that the multiplicity changes by nwhen crossing a face of Pwith
multiplicity n; we can do this consistently, since the intersection of any loop avoiding
Kwith Pis zero (this is only well defined up to a global additive constant, but
this will not matter). This yields a (n +1)chain Qin ×{T}whose boundary is
a chain in Kalong with the part of Pthat is disjoint from K.NowP−∂Q has
∂(P−∂Q) =Cand is supported in K. As such, P−∂Q is disjoint from M(t ).
The result follows.
Acknowledgements We are grateful to Richard Bamler, Costante Bellettini, Robert Haslhofer, Or Her-
shkovits, Daren Cheng, Ciprian Manolescu, Leon Simon, and Brian White for useful conversations related
to this paper as well as to the anonymous referees for their careful reading and many helpful suggestions.
For the purpose of open access, the authors have applied a Creative Commons Attribution (CC-BY) license
to any Author Accepted Manuscript version arising from this submission.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
216 O. Chodosh et al.
Funding O.C. was partially supported by a Terman Fellowship, a Sloan Fellowship, and NSF grants DMS-
1811059, DMS-2016403, and DMS-2304432. He would also like to acknowledge the MATRIX Institute
for its hospitality during the time which some of this article was completed. K.C. was supported by the
KIAS Individual Grant MG078902, a POSCO Science Fellowship, an Asian Young Scientist Fellowship,
and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)
(RS-2023-00219980). C.M. was supported by the NSF grant DMS-1905165. F.S. was supported by a
Leverhulme Trust Research Project Grant RPG-2016-174.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-
mons licence, and indicate if changes were made. The images or other third party material in this article
are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly
from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/
4.0/.
References
1. Angenent, S.B.: Shrinking doughnuts. In: Nonlinear Diffusion Equations and Their Equilibrium
States, 3 (Gregynog, 1989). Progr. Nonlinear Differential Equations Appl., vol. 7, pp. 21–38.
Birkhäuser, Boston (1992)
2. Angenent, S., Chopp, D.L., Ilmanen, T.: A computed example of nonuniqueness of mean curvature
flow in R3. Commun. Partial Differ. Equ. 20(11–12), 1937–1958 (1995)
3. Angenent, S., Daskalopoulos, P., Sesum, N.: Unique asymptotics of ancient convex mean curvature
flow solutions. J. Differ. Geom. 111(3), 381–455 (2019)
4. Angenent, S., Daskalopoulos, P., Sesum, N.: Uniqueness of two-convex closed ancient solutions to
the mean curvature flow. Ann. Math. (2) 192(2), 353–436 (2020)
5. Angenent, S., Brendle, S., Daskalopoulos, P., Šešum, N.: Unique asymptotics of compact ancient
solutions to three-dimensional Ricci flow. Commun. Pure Appl. Math. 75(5), 1032–1073 (2022)
6. Bamler, R.H., Kleiner, B.: On the multiplicity one conjecture for mean curvature flows of surfaces
(2023). https://arxiv.org/pdf/2312.02106.pdf
7. Bernstein, J., Wang, L.: A sharp lower bound for the entropy of closed hypersurfaces up to dimension
six. Invent. Math. 206(3), 601–627 (2016)
8. Bernstein, J., Wang, L.: A topological property of asymptotically conical self-shrinkers of small
entropy. Duke Math. J. 166(3), 403–435 (2017)
9. Bernstein, J., Wang, L.: Hausdorff stability of the round two-sphere under small perturbations of the
entropy. Math. Res. Lett. 25(2), 347–365 (2018)
10. Bernstein, J., Wang, L.: An integer degree for asymptotically conical self-expanders (2018). https://
arxiv.org/abs/1807.06494
11. Bernstein, J., Wang, L.: Topology of closed hypersurfaces of small entropy. Geom. Topol. 22(2),
1109–1141 (2018)
12. Bernstein, J., Wang, L.: Smooth compactness for spaces of asymptotically conical self-expanders of
mean curvature flow. Int. Math. Res. Not. 12, 9016–9044 (2021)
13. Bernstein, J., Wang, L.: The space of asymptotically conical self-expanders of mean curvature flow.
Math. Ann. 380(1–2), 175–230 (2021)
14. Bernstein, J., Wang, L.: Closed hypersurfaces of low entropy in R4are isotopically trivial. Duke
Math. J. 171(7), 1531–1558 (2022)
15. Bernstein, J., Wang, L.: Relative expander entropy in the presence of a two-sided obstacle and appli-
cations. Adv. Math. 399, Paper No. 108284, 48 (2022)
16. Bernstein, J., Wang, L.: Topological uniqueness for self-expanders of small entropy. Camb. J. Math.
10(4), 785–833 (2022)
17. Bourni, T., Langford, M., Mramor, A.: On the construction of closed nonconvex nonsoliton ancient
mean curvature flows. Int. Math. Res. Not. 1, 757–768 (2021)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Mean curvature flow with generic initial data 217
18. Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Thesis (Ph.D.)–Princeton University.
ProQuest LLC, Ann Arbor (1975)
19. Brendle, S.: A sharp bound for the inscribed radius under mean curvature flow. Invent. Math. 202(1),
217–237 (2015)
20. Brendle, S.: Embedded self-similar shrinkers of genus 0. Ann. Math. (2) 183(2), 715–728 (2016)
21. Brendle, S.: Ancient solutions to the Ricci flow in dimension 3. Acta Math. 225(1), 1–102 (2020)
22. Brendle, S., Choi, K.: Uniqueness of convex ancient solutions to mean curvature flow in R3.Invent.
Math. 217(1), 35–76 (2019)
23. Brendle, S., Choi, K.: Uniqueness of convex ancient solutions to mean curvature flow in higher
dimensions. Geom. Topol. 25(5), 2195–2234 (2021)
24. Brendle, S., Huisken, G.: Mean curvature flow with surgery of mean convex surfaces in R3.Invent.
Math. 203(2), 615–654 (2016)
25. Brendle, S., Kapouleas, N.: Gluing Eguchi-Hanson metrics and a question of Page. Commun. Pure
Appl. Math. 70(7), 1366–1401 (2017)
26. Brendle, S., Huisken, G., Sinestrari, C.: Ancient solutions to the Ricci flow with pinched curvature.
Duke Math. J. 158(3), 537–551 (2011)
27. Brendle, S., Daskalopoulos, P., Sesum, N.: Uniqueness of compact ancient solutions to three-
dimensional Ricci flow. Invent. Math. 226(2), 579–651 (2021)
28. Buzano, R., Nguyen, H.T., Schulz, M.B.: Noncompact self-shrinkers for mean curvature flow with
arbitrary genus (2021). https://arxiv.org/abs/2110.06027
29. Cheeger, J., Haslhofer, R., Naber, A.: Quantitative stratification and the regularity of mean curvature
flow. Geom. Funct. Anal. 23(3), 828–847 (2013)
30. Chen, B.-L., Yin, L.: Uniqueness and pseudolocality theorems of the mean curvature flow. Commun.
Anal. Geom. 15(3), 435–490 (2007)
31. Chodosh, O., Schulze, F.: Uniqueness of asymptotically conical tangent flows. Duke Math. J.
170(16), 3601–3657 (2021)
32. Chodosh, O., Choi, K., Mantoulidis, C., Schulze, F.: Mean curvature flow with generic low-entropy
initial data. Duke Math. J. (2021, to appear). https://arxiv.org/pdf/2102.11978.pdf
33. Chodosh, O., Eichmair, M., Shi, Y., Yu, H.: Isoperimetry, scalar curvature, and mass in asymptoti-
cally flat Riemannian 3-manifolds. Commun. Pure Appl. Math. 74(4), 865–905 (2021)
34. Chodosh, O., Choi, K., Schulze, F.: Mean curvature flow with generic initial data II (2023). https://
arxiv.org/pdf/2302.08409.pdf
35. Chodosh, O., Daniels-Holgate, J.M., Schulze, F.: Mean curvature flow from conical singularities.
(2023). https://arxiv.org/pdf/2312.00759.pdf
36. Chodosh, O., Mantoulidis, C., Schulze, F.: Generic regularity for minimizing hypersurfaces in di-
mensions 9 and 10 (2023). https://arxiv.org/pdf/2302.02253.pdf
37. Chodosh, O., Mantoulidis, C., Schulze, F.: Mean curvature flow with generic low-entropy initial data
II (2023). https://arxiv.org/pdf/2309.03856.pdf
38. Choi, K., Mantoulidis, C.: Ancient gradient flows of elliptic functionals and Morse index. Am. J.
Math. 144(2), 541–573 (2022)
39. Choi, K., Haslhofer, R., Hershkovits, O.: Ancient low-entropy flows, mean-convex neighborhoods,
and uniqueness. Acta Math. 228(2), 217–301 (2022)
40. Choi, K., Haslhofer, R., Hershkovits, O., White, B.: Ancient asymptotically cylindrical flows and
applications. Invent. Math. 229(1), 139–241 (2022)
41. Chopp, D.L.: Computation of self-similar solutions for mean curvature flow. Exp. Math. 3(1), 1–15
(1994)
42. Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I: generic singularities. Ann. Math.
(2) 175(2), 755–833 (2012)
43. Colding, T.H., Minicozzi II, W.P.: Smooth compactness of self-shrinkers. Comment. Math. Helv.
87(2), 463–475 (2012)
44. Colding, T.H., Minicozzi II, W.P.: Uniqueness of blowups and łojasiewicz inequalities. Ann. Math.
(2) 182(1), 221–285 (2015)
45. Colding, T.H., Minicozzi II, W.P.: The singular set of mean curvature flow with generic singularities.
Invent. Math. 204(2), 443–471 (2016)
46. Colding, T.H., Ilmanen, T., Minicozzi, W.P. II, White, B.: The round sphere minimizes entropy
among closed self-shrinkers. J. Differ. Geom. 95(1), 53–69 (2013)
47. Colding, T.H., Ilmanen, T., Minicozzi II, W.P.: Rigidity of generic singularities of mean curvature
flow. Publ. Math. Inst. Hautes Études Sci. 121, 363–382 (2015)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
218 O. Chodosh et al.
48. Daskalopoulos, P., Hamilton, R., Sesum, N.: Classification of compact ancient solutions to the curve
shortening flow. J. Differ. Geom. 84(3), 455–464 (2010)
49. Daskalopoulos, P., Hamilton, R., Sesum, N.: Classification of ancient compact solutions to the Ricci
flow on surfaces. J. Differ. Geom. 91(2), 171–214 (2012)
50. Daskalopoulos, P., del Pino, M., Sesum, N.: Type II ancient compact solutions to the Yamabe flow.
J. Reine Angew. Math. 738, 1–71 (2018)
51. De Giorgi, E., Stampacchia, G.: Sulle singolarità eliminabili delle ipersuperficie minimali. Atti Ac-
cad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 8(38), 352–357 (1965)
52. Ecker, K.: Logarithmic Sobolev inequalities on submanifolds of Euclidean space. J. Reine Angew.
Math. 522, 105–118 (2000)
53. Ecker, K.: Regularity Theory for Mean Curvature Flow. Progress in Nonlinear Differential Equations
and Their Applications, vol. 57. Birkhäuser Boston, Boston (2004)
54. Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. (2) 130(3), 453–471
(1989)
55. Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math.
105(3), 547–569 (1991)
56. Edelen, N., Haslhofer, R., Ivaki, M.N., Zhu, J.J.: Mean convex mean curvature flow withfree bound-
ary. Commun. Pure Appl. Math. 75(4), 767–817 (2022)
57. Evans, L.C.: Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, vol. 19. Am.
Math. Soc., Providence (2010)
58. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33(3), 635–681
(1991)
59. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. IV. J. Geom. Anal. 5(1), 77–114
(1995)
60. Hardt, R., Simon, L.: Area minimizing hypersurfaces with isolated singularities. J. Reine Angew.
Math. 362, 102–129 (1985)
61. Haslhofer, R., Hershkovits, O.: Ancient solutions of the mean curvature flow. Commun. Anal. Geom.
24(3), 593–604 (2016)
62. Haslhofer, R., Hershkovits, O.: Singularities of mean convex level set flow in general ambient man-
ifolds. Adv. Math. 329, 1137–1155 (2018)
63. Haslhofer, R., Kleiner, B.: Mean curvature flow of mean convex hypersurfaces. Commun. Pure Appl.
Math. 70(3), 511–546 (2017)
64. Haslhofer, R., Kleiner, B.: Mean curvature flow with surgery. Duke Math. J. 166(9), 1591–1626
(2017)
65. Hirsch, M.W.: Differential Topology. Graduate Texts in Mathematics, vol. 33. Springer, New York
(1976)
66. Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1),
237–266 (1984)
67. Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom.
31(1), 285–299 (1990)
68. Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature. In: Differential
Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990). Proc. Sympos.
Pure Math., vol. 54, pp. 175–191. Am. Math. Soc., Providence (1993)
69. Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean
convex surfaces. Acta Math. 183(1), 45–70 (1999)
70. Huisken, G., Sinestrari, C.: Mean curvature flow singularities for mean convex surfaces. Calc. Var.
Partial Differ. Equ. 8(1), 1–14 (1999)
71. Huisken, G., Sinestrari, C.: Mean curvature flow with surgeries of two-convex hypersurfaces. Invent.
Math. 175(1), 137–221 (2009)
72. Huisken, G., Sinestrari, C.: Convex ancient solutions of the mean curvature flow. J. Differ. Geom.
101(2), 267–287 (2015)
73. Ilmanen, T.: The level-set flow on a manifold. In: Differential Geometry: Partial Differential Equa-
tions on Manifolds (Los Angeles, CA, 1990). Proc. Sympos. Pure Math., vol. 54, pp. 193–204. Am.
Math. Soc., Providence (1993)
74. Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Am.
Math. Soc. 108(520), x+90 (1994)
75. Ilmanen, T.: Lectures on mean curvature flow and related equations (Trieste notes) (1995). https://
people.math.ethz.ch/~ilmanen/papers/notes.ps
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Mean curvature flow with generic initial data 219
76. Ilmanen, T.: Singularities of mean curvature flow of surfaces (1995). https://people.math.ethz.ch/
~ilmanen/papers/sing.ps
77. Ilmanen, T.: A strong maximum principle for singular minimal hypersurfaces. Calc. Var. Partial
Differ. Equ. 4(5), 443–467 (1996)
78. Ilmanen, T.: Problems in mean curvature flow (2003). https://people.math.ethz.ch/~ilmanen/classes/
eil03/problems03.ps
79. Ilmanen, T., White, B.: Sharp lower bounds on density for area-minimizing cones. Camb. J. Math.
3(1–2), 1–18 (2015)
80. Ilmanen, T., Neves, A., Schulze, F.: On short time existence for the planar network flow. J. Differ.
Geom. 111(1), 39–89 (2019)
81. Kapouleas, N., Kleene, S.J., Møller, N.M.: Mean curvature self-shrinkers of high genus: non-
compact examples. J. Reine Angew. Math. 739, 1–39 (2018)
82. Ketover, D.: Self-shrinking platonic solids (2016). https://arxiv.org/abs/1602.07271
83. Kleiner, B., Lott, J.: Singular Ricci flows I. Acta Math. 219(1), 65–134 (2017)
84. Knerr, B.F.: Parabolic interior Schauder estimates by the maximum principle. Arch. Ration. Mech.
Anal. 75(1), 51–58 (1980/81)
85. Lin, L.: Mean curvature flow of star-shaped hypersurfaces. Commun. Anal. Geom. 28(6), 1315–1336
(2020)
86. Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. (2) 179(2),
683–782 (2014)
87. Nguyen, X.H.: Construction of complete embedded self-similar surfaces under mean curvature flow,
Part III. Duke Math. J. 163(11), 2023–2056 (2014)
88. Or, H., White, B.: Sharp entropy bounds for self-shrinkers in mean curvature flow. Geom. Topol.
23(3), 1611–1619 (2019)
89. Or, H., White, B.: Nonfattening of mean curvature flow at singularities of mean convex type. Com-
mun. Pure Appl. Math. 73(3), 558–580 (2020)
90. Or, H., White, B.: Avoidance for set-theoretic solutions of mean-curvature-type flows. Commun.
Anal. Geom. 31(1), 31–67 (2023)
91. Schulze, F.: Uniqueness of compact tangent flows in mean curvature flow. J. Reine Angew. Math.
690, 163–172 (2014)
92. Schulze, F., White, B.: A local regularity theorem for mean curvature flow with triple edges. J. Reine
Angew. Math. 758, 281–305 (2020)
93. Sesum, N.: Rate of convergence of the mean curvature flow. Commun. Pure Appl. Math. 61(4),
464–485 (2008)
94. Simon, L.: Schauder estimates by scaling. Calc. Var. Partial Differ. Equ. 5(5), 391–407 (1997)
95. Smale, N.: Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional
manifolds. Commun. Anal. Geom. 1(2), 217–228 (1993)
96. Smoczyk, K.: Starshaped hypersurfaces and the mean curvature flow. Manuscr. Math. 95(2),
225–236 (1998)
97. Solomon, B., White, B.: A strong maximum principle for varifolds that are stationary with respect
to even parametric elliptic functionals. Indiana Univ. Math. J. 38(3), 683–691 (1989)
98. Sun, A.: Local entropy and generic multiplicity one singularities of mean curvature flow of surfaces.
J. Differ. Geom. 124(1), 169–198 (2023)
99. Sun, A., Wang, Z.: Compactness of self-shrinkers in R3with fixed genus. Adv. Math. 367, 107110
(2020)
100. Wang, X.-J.: Convex solutions to the mean curvature flow. Ann. Math. (2) 173(3), 1185–1239 (2011)
101. Wang, L.: Uniqueness of self-similar shrinkers with asymptotically conical ends. J. Am. Math. Soc.
27(3), 613–638 (2014)
102. Wang, L.: Asymptotic structure of self-shrinkers (2016). https://arxiv.org/abs/1610.04904
103. White, B.: The topology of hypersurfaces moving by mean curvature. Commun. Anal. Geom. 3(1–2),
317–333 (1995)
104. White, B.: Stratification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine
Angew. Math. 488, 1–35 (1997)
105. White, B.: The size of the singular set in mean curvature flow of mean-convex sets. J. Am. Math.
Soc. 13(3), 665–695 (2000)
106. White, B.: Evolution of curves and surfaces by mean curvature. In: Proceedings of the International
Congress of Mathematicians, Vol. I (Beijing, 2002), pp. 525–538. Higher Ed. Press, Beijing (2002)
107. White, B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Am. Math.
Soc. 16(1), 123–138 (2003)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
220 O. Chodosh et al.
108. White, B.: A local regularity theorem for mean curvature flow. Ann. Math. (2) 161(3), 1487–1519
(2005)
109. White, B.: Currents and flat chains associated to varifolds, with an application to mean curvature
flow. Duke Math. J. 148(1), 41–62 (2009)
110. White, B.: Topological change in mean convex mean curvature flow. Invent. Math. 191(3), 501–525
(2013)
111. White, B.: Subsequent singularities in mean-convex mean curvature flow. Calc. Var. Partial Differ.
Equ. 54(2), 1457–1468 (2015)
112. White, B.: Mean curvature flow with boundary. Ars Inven. Anal. 2021, Paper No. 4, 43 (2021)
113. Zhu, J.J.: On the entropy of closed hypersurfaces and singular self-shrinkers. J. Differ. Geom. 114(3),
551–593 (2020)
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