Mean curvature flow from conical singularities

Abstract

We prove Ilmanen’s resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface Mn⊂Rn+1M^{n}\subset \mathbb{R}^{n+1}, 2≤n≤62\leq n\leq 6, with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens.

Full text

Inventiones mathematicae (2024) 238:1041–1066
https://doi.org/10.1007/s00222-024-01296-8
Mean curvature flow from conical singularities
Otis Chodosh1·J.M. Daniels-Holgate2·Felix Schulze3
Received: 13 December 2023 / Accepted: 27 October 2024 / Published online: 8 November 2024
© The Author(s) 2024
Abstract
We prove Ilmanen’s resolution of point singularities conjecture by establishing short-
time smoothness of the level set flow of a smooth hypersurface with isolated conical
singularities. This shows how the mean curvature flow evolves through asymptoti-
cally conical singularities. Precisely, we prove that the level set flow of a smooth
hypersurface Mn⊂Rn+1,2≤n≤6, with an isolated conical singularity is modeled
on the level set flow of the cone. In particular, the flow fattens (instantaneously) if
and only if the level set flow of the cone fattens.
1Introduction
A family of smooth hypersurfaces M(t) is a mean curvature flow if
(∂
∂t x)⊥=HM(t)(x),
where HM(t)(x)is the mean curvature vector of M(t) at x. Mean curvature flow is the
gradient flow of area. We recall that the mean curvature flow, M(t), from a smooth,
compact hypersurface M(0)⊂Rn+1is guaranteed to become singular in finite time,
moreover, well-posedness and regularity of the flow can break down after the onset
of certain singularities (cf. [50]).
F. Schulze
felix.schulze@warwick.ac.uk
O. Chodosh
ochodosh@stanford.edu
J.M. Daniels-Holgate
joshua.daniels-holgate@mail.huji.ac.il
1Department of Mathematics, Stanford University, Bldg. 380, Stanford, CA 94305, USA
2Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of
Jerusalem, Givat Ram. Jerusalem, 9190401, Israel
3Department of Mathematics, Zeeman Building, University of Warwick, Gibbet Hill Road,
Coventry CV4 7AL, UK
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1042 O. Chodosh et al.
In the present article, we quantify the short-time regularity and well-posedness of
the level set flow from a smooth compact hypersurface with an isolated singularity1
modeled on any smooth cone C. Recalling [20], such hypersurfaces can appear as the
singular time-slice of a flow encountering a singularity modeled on an asymptotically
conical self-shrinker. Our results hence demonstrate how one can flow through such
a singularity.
Before stating our results, we recall that the level set flow (cf. [19,32,37,44])
ofaclosedsetXis the unique maximal assignment of closed sets t→ Ft(X) with
F0(X) =X, such that Ft(X) avoids smooth flows (see Sect. 2.1). If the level set flow
Ft(X) develops an interior at t=T, we say that the flow fattens at time T.
Our main results can be stated as follows:
Theorem 1.1 (Fattening dichotomy) For 2≤n≤6, suppose that Mn⊂Rn+1is a
smooth hypersurface with an isolated conical singularity modeled on a smooth cone
C.Then the level set flow of Mfattens instantly if and only if the level-set flow from
the cone Cfattens.
Fattening of Cimplies fattening of Mis proven in Theorem 4.1, whilst non-
fattening of Cimplies short-time non-fattening of Mcan be found in Corollary 6.3.
A fortiori, Theorem 1.1 is a consequence of the following results (more precisely
Theorem 1.2 and Theorem 1.4), which give a precise description of the level-set flow
near the conical singularity of M.
Theorem 1.2 (Structure theorem for the level set flow) Fo r 2≤n≤6, suppose that
Mn⊂Rn+1is a smooth hypersurface with an isolated conical singularity modeled on
a smooth cone Cat 0.Then,there is a T>0such that the outermost mean curvature
flows of Mare smooth for t∈(0,T).Moreover,t−1/2Ft(M ) converges in the local
Hausdorff sense to F1(C)as t0.
We provide a refinement of this statement below, which, in aggregate with the
aforementioned work [20], can be considered as a canonical neighbourhood theorem
for asymptotically conical singularities. Before stating this result, we provide a brief
exposition of the Hershkovits–White framework applicable to the present context.
(See Sect. 2.3 for a rigorous discussion.)
We first consider the compact case, illustrated in Fig. 1. Recall that the outer flow,
Mis the space-time boundary of the level set flow Ft(V ) of the interior Vof M.
Similarly the inner flow Mis the space-time boundary of the level set flow Ft(V )
of the exterior Vof M. Turning to the cone Cas illustrated in Fig. 2, we note dilation
invariance and uniqueness of the level set flow yields Ft(C)=√tF1(C). Denote W
and Wthe interior and exterior of the cone Cand assume we have chosen these conis-
tently with the interior and exterior of M.Let:= ∂F1(W )and :=∂F1(W ) and
observe that ∂Ft(W ) =√t,∂Ft(W )=√t. Note, when 2 ≤n≤6, ,will
1For simplicity of notation we only consider a single singularity, but everything here would generalize
easily to the case of finitely many isolated singularities, each modeled on a smooth cone.
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MCF from conical singularities 1043
Fig. 1 Left: A hypersurface Mwith isolated conical singularity, with interior Vand exterior V. Right:
The level set flow of each at time t. (Color figure online.)
Fig. 2 Left: The initial cone Cwith interior Wand exterior W. Right: The level set flow of each region
at time t=1. (Color figure online.)
be smooth (this is the source of the dimension restriction above; for n>6, the above
theorems continue to hold if we impose the additional condition that the outermost
expanders for Care smooth).
In the sequel we will refer to M(t),M(t) as outermost flows and ,as the
outermost expanders. The next result shows that the outermost expanders approxi-
mate the outermost flows.
Theorem 1.3 (Canonical neighbourhood theorem for outermost flows) For 2≤n≤
6, suppose that Mn⊂Rn+1is a smooth hypersurface with an isolated conical sin-
gularity at 0.Assume the conical singularity is modeled on a smooth cone Cwith
outermost expanders ,labeled as above.Then,t−1/2M(t) (resp.t−1/2M(t ))
converges to (resp.)locally smoothly as t0.
Theorem 1.3 resolves the “resolution of point singularities” conjecture of Ilmanen
[40, Problem 16]. Smoothness can be found in Corollary 4.2 and the forward blow-up
statement (including the convergence of the outermost Brakke flows to the outermost
expanders) can be found in Theorem 4.1.
Note that if Cdoes not fatten then =and Theorem 1.2 trivially holds. In
particular, any flows starting from Mare smooth for a short time and modeled on
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1044 O. Chodosh et al.
the unique expander asymptotic to C. This implies that two such flows separate like
o(t1/2), but this does not a priori imply there is only one such flow. This is the content
of our final result.
Theorem 1.4 (Uniqueness) Fo r 2≤n≤6, suppose that Mn⊂Rn+1is a smooth
hypersurface with an isolated conical singularity at 0,modeled on a smooth cone C
which does not fatten.Then,there is T>0such that the outermost flows of Magree
(and are smooth)for t∈(0,T).Especially,the evolution of Mis unique on this time
interval.
Smoothness follows again from Corollary 4.2 and for uniqueness see Corollary
6.3.
Remark 1.5 As a consequence of the work of Brendle [14] any asymptotically coni-
cal shrinker must have non-zero genus and by work of Ilmanen–White [38,p.21]
the inner and outer expanders are topological planes. Combined with Chodosh–
Schulze [20, Corollary 1.2], the results presented in this work demonstrate strict
genus drop through any isolated conical singularities that form in a multiplicity one
flow. We note that the full “strict genus monotonicity conjecture” of Ilmanen [40,
Problem 13] at non-generic singularities (for outermost flows) was recently resolved
by Bamler–Kleiner [6] by combining their resolution of Ilmanen’s multiplicity one
conjecture with the strict genus drop results for one-sided perturbations of Chodosh–
Choi–Schulze–Mantoulidis [21,22].
1.1 Related work
The study of fattening and non-fattening of conical singularities has received consid-
erable attention. In particular, in their first work on the level set flow, Evans–Spruck
already observed [32, §8.2] that the cone C:={xy =0}⊂R2and a figure eight will
fatten. Note that a figure eight is a smooth curve in R2with an isolated conical singu-
larity modeled on the cone Cin the terminology of this paper (and our results would
apply without change to this setting). Fattening has been subsequently studied by
many authors, see [1,3,4,27,31,33,35,37,38,47,48,50] for a non-exhaustive
list.
More recently, Hershkovits–White [43] introduced a powerful framework for
analysing the level set flow, which they applied to show non-fattening through mean-
convex singularities. Combining their work with the resolution of the mean-convex
neighborhood conjecture by Choi–Haslhofer–Herskovits [24](cf.[25]), it follows
that fattening does not occur if all singularities are either round cylinders of the
form Sn−1×Ror round spheres Sn. We also draw attention to the recent studies
of asymptotically conical expanders by Deruelle–Schulze [29] and Bernstein–Wang
[7,8,10–13]. In particular, Bernstein–Wang have used these results to prove a low-
entropy Schoenflies theorem [9](cf.[22,23,26]) and have announced applications
to the study of low-entropy cones. See also the work of Chen [15–17].
Finally, we note that the question of evolving a Ricci flow through a singularity
modelled on the evolution of an asymptotically conical gradient shrinking soliton is
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MCF from conical singularities 1045
also of considerable interest (but we note that the analogues of Theorem 1.1 and 1.3
and the resolution of point singularities are not understood in general). In particu-
lar, expanders have been studied in [5,28,30,45] and flows have been constructed
out of initial Riemannian manifolds with isolated conical singularities modeled on
non-negatively curved cones over spheres [34]. Moreover, “fattening” of the cone at
infinity of a shrinking gradient Ricci soliton has been constructed in [2].
1.2 Strategy of proof
Optimistically, one might hope that the resolution of a conical singularity is always
modeled on expanders, just as tangent flows are always modeled on self-shrinkers.
Indeed, one might expect a forward monotonicity formula would control the forward
blow-ups (the (subsequential) weak limits of λ2M(λ−1t) as λ→∞) but there appear
to be serious issues to make this rigorous in the setting of isolated conical singular-
ities (cf. [38, p. 25]). We do note that in the setting of flows coming out of cones,
Bernstein–Wang have obtained a version of forwards monotonicity [11] (generaliz-
ing to the dynamical setting the relative expander entropy of Deruelle–Schulze [29])
and Chen [15] has constructed non-self-expanding flows from cones. However, it re-
mains unclear if/how monotonicity based methods could prove that forward blowups
of outermost flows are outermost expanders (or even that they are smooth).
In this article we take a completely different approach (avoiding forwards mono-
tonicity entirely). Instead, we find barriers that push the outermost flows onto the out-
ermost expanders in the forward blowup limit. A closely related construction proves
uniqueness of two flows with the same outermost expander blowup limit. The con-
struction of these barriers combines two key spectral properties of an outermost ex-
pander :
(1) The outermost expander minimizes weighted area to the outside, so the linearized
expander operator (cf. (2.4)) is non-negative L≥0. In particular, there is a
positive eigenfunction φ3Ron ∩B3R(0)with positive eigenvalue μ3R>0.
(2) The outermost expander is the one-sided limit of expanders asymptotic to nearby
cones, which yields a positive Jacobi field Lv=0 with vgrowing linearly at
infinity.
The “interior” barrier is then formed by taking the graph over of a small multiple
of f:=v+αφ3R. Because Lf=−αμφ3Rthis can be seen to be a strict barrier in
B2R(0), pushing (rescaled) mean curvature flows towards .
To prove that the flow fattens if the cone fattens (Theorem 4.1), we can weld (in the
sense of Meeks–Yau [42]) the graph of fto the graph of a constant function hover 
to obtain a global barrier sover (note that Lh=(|A|2−1
2)h is <0 outside of a
sufficiently large compact set). (See Proposition 3.4.) Now, the key observation is that
the forward blowups of the outermost flow will lie below soutside of a sufficently
large set, since the forward blowups must lie in the level set flow of the cone (which
decay towards the cone) while shas height ∼sh over the cone near infinity (see
Claim 4.2).
In particular, this proves that the outermost flows have forward blowup at 0equal
to the outermost flows of the cone C. To prove that the flow does not fatten if the
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1046 O. Chodosh et al.
Fig. 3 The barrier construction
to prove uniqueness of flows
with the same (outermost)
expander as forward blowup
at 0. (Color figure online.)
cone does not fatten, it thus suffices to consider two flows M1(t),M2(t ) which have
forwards blowup given by the same outermost expander. We construct a barrier in
this situation by welding (in the sense of Meeks–Yau [42]) the graph of ±sf (denoted
±
close,s (t)) to the normal graph of sh√tover M1(t) \B√tR(0)(denoted ±
far,s (t)).
The barriers then pinch M2towards M1from above and below as s→0 proving
uniqueness. This can be seen in Fig. 3.
1.3 Organization
In Sect. 2we collect some preliminary definitions and facts to be used later. In Sect. 3
we construct barriers graphical over the expander and then use these barriers to prove
that the level set flow is locally modeled on the level set flow of the cone in Sect. 4.
In Sect. 5we construct global barriers over a flow that’s modeled on an outermost
expander near the conical singularity and then use these to prove uniqueness of such
flows in Sect. 6. Finally, we collect some results about graphs over expanders in the
Appendix.
2Preliminaries
In this section we collect some preliminary definitions, conventions, and results.
2.1 Spacetime and the level set flow
We define the time map t:Rn+1×R→Rto be the projection t(x,t)=t.ForE⊂
Rn+1×Rwe will write E(t) := E∩t−1(t). The knowledge of E(t) for all tis the
same thing as knowing E, so we will often ignore the distinction.
For a compact n-manifold M(possibly with boundary), we consider f:M×
[a,b]→Rn+1so that (i) fis continuous (ii) fis smooth on (M \∂M) ×(a , b](iii)
f|M×{t}is injective for each t∈[a, b]and (iii) t→f(M\∂M, t) is flowing by mean
curvature flow. In this case we call
M:=∪t∈[a,b]f(M,t)×{t}⊂Rn+1×R
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MCF from conical singularities 1047
aclassical mean curvature flow and define the heat boundary of Mby
∂M:=f(M,a)∪f(∂M,[a,b]).
Classical flows that intersect must intersect in a point that belongs to at least one of
their heat boundaries (cf. [49, Lemma 3.1]).
For ⊂Rn+1×[0,∞),M⊂Rn+1×Ris a weak set flow (generated by )
if M(0)=(0)and if Mis a classical flow with ∂Mdisjoint from Mand M
disjoint from then Mis disjoint from M. There may be more than one weak set
flow generated by .
The biggest such flow is called the level set flow, which can be constructed as
follows: For ⊂Rn+1×[0,∞)as above, we set
W0:={(x,0):(x,0)/∈}
and then let Wk+1denote the union of all classical flows Mwith Mdisjoint from
and ∂M⊂Wk.Thelevel set flow generated by is then defined by
M:=(Rn+1×[0,∞)) \∪kWk⊂Rn+1×[0,∞).
See [32,37,49]. If ⊂Rn+1×{0}, we will write Ft() :=M(t ) for the time tslice
of the corresponding level set flow.
Fix ⊂Rn+1closed. We say that the level set flow of is non-fattening if Ft()
has no interior for each t≥0. This condition holds generically for compact ⊂
Rn+1, namely if u0is a continuous function with compact level sets u−1
0(s) then
the level set flow of u−1
0(s) fattens for at most countably many values of s,see[37,
§11.3-4].
2.2 Integral Brakke flows
An (n-dimensional2) integral Brakke flow in Rn+1is a 1-parameter family of Radon
measures (μ(t))t∈Iso that
(1) For almost every t∈Ithere is an integral n-dimensional varifold V(t) with
μ(t) =μV(t) and so that V(t)has locally bounded first variation and mean cur-
vature Horthogonal to Tan(V (t ), ·)almost everywhere.
(2) For every bounded interval [t1,t
2]⊂Iand K⊂Rn+1compact, we have
t2
t1K
(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(t2)−f(·,t
1)dμ(t1)≤t2
t1(−|H|2f+H·∇f+∂f
∂t )dμ(t)dt.
2Of course one can consider k-dimensional flows in Rn+1butwewillneverdosointhispaper,sowe
will often omit the “n-dimensionality” and implicitly assume that all Brakke flows are flows of “hypersur-
faces.”.
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1048 O. Chodosh et al.
We will sometimes write Mto represent a Brakke flow.
We define the support of M=(μ(t))tto be ∪tsupp μ(t) ×{t}⊂Rn+1×R.Itis
useful to recall that the support a Brakke flow (with t∈[0,∞)) is a weak set flow
(generated by supp μ(0))[37, 10.5].
We say that a sequence of integral Brakke flows Miconverges to another integral
Brakke flow M(written MiM)ifμi(t) weakly converges to μ(t) for all tand
for almost every t, after passing to a further subsequence depending on t, the asso-
ciated integral varifolds converge Vi(t) →V(t). (Recall that if Miis a sequence of
integral Brakke flows with uniform local mass bounds then a subsequence converges
to an integral Brakke flow [37, §7].)
For a Brakke flow Mand λ>0 we write Dλ(M)for the “dilated” Brakke flows
with measures satisfying μλ(t)(A) =λnμ(λ−2t)(λ−1A).
2.3 The inner/outer flows of a level set flow
We collect results of [43] on weak set flows and outermost flows and show that they
are also applicable (with minor modifications) to the flow of more general initial data.
Proposition 2.1 ([43, Proposition A.3]) Suppose that Fis a 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 assume that Fis the closure of its interior in Rn+1(we will
call3such a set Fadmissible). 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 2.1,weset:
M(t) :={(x,t)⊂Rn+1:x∈∂M},
M(t) :={(x,t)⊂Rn+1:x∈∂M}.
(Here ∂M,∂Mare the relative boundaries of M,Mas subsets of Rn+1×R+).
We call
t→M(t), t → M(t )
the outer and inner flows of M:=∂F. By Proposition 2.1,M(t),M(t) are contained
in the level set flow generated by M. Furthermore,
M(t) =lim
τt∂F(τ)
3Note that this slightly extends the definition in [43], 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 [43].
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MCF from conical singularities 1049
for all t>0, and M(t) =∂F(t) for all but countably many t. See [43, Theorems
B.5, C.10]. Note that [43, Theorems B.5] directly carries over to M=∂F where F
is admissible.
We will say that an admissible set Fis smoothable, if the following holds: There
exist compact regions FFiwith smooth boundaries such that
(1) For each i,Fi+1is contained in the interior of Fi.
(2) ∩Fi=intF.
(3) HnMis a Radon measure and Hn∂Fi→HnM.
(4) There is >0 so that for any p∈Rn+1and ρ>0 it holds that |∂Fi∩Bρ(p)|≤
ρ n.
By perturbing Fislightly, we can also assume that
(5) the level set flow of ∂Finever fattens.
Choose integral Brakke flows t→ μi(t) starting from μi(0)=Hn∂Fivia elliptic
regularization. Assume that μilimits to t→μ(t ) in the sense of Brakke flows. Note
that the flow t∈[0,∞)→μ(t) is an integral, unit-regular Brakke flow with μ(0)=
HnM
We do the same hold with Freplacing Fand so on. We then directly generalize
[43, Theorems B.6, B.8]. The proof extends verbatim.
Proposition 2.2 Assume Fis admissible and smoothable with M=∂F The Brakke
flow t→ μ(t) has spacetime support equal to the spacetime set swept out by t∈
[0,∞)→ M(t),where t→ M(t) is the outer flow of M.More precisely,for t>0,
the Gaussian density of the flow μ(·)at (x,t) is >0if and only if x∈M(t).The
analogous statement holds for the inner flow t→M(t) of M.
2.4 Density, Huisken’s monotonicity, and entropy
For X0=(x0,t
0)∈Rn+1×Rwe consider the (n-dimensional) backwards heat kernel
based at X0:
ρX0(x,t):=(4π(t0−t))−n
2exp −|x−x0|2
4(t0−t)(2.1)
for x∈Rn+1,t<t
0.ForMa Brakke flow defined on [T0,∞),t0>T
0and 0 <r ≤
√T0−t0,weset
M(X0,r):=ρX0(x,t
0−r2)dμ(t0−r2).
Huisken’s monotonicity formula [36,39] implies that r→ M(X0,r) is non-
decreasing (and constant only for a shrinking self-shrinker centered at X0). In partic-
ular we can define the density of Mat such X0by
M(X0):= lim
r0M(X0,r).
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1050 O. Chodosh et al.
We call an integral Brakke flow Munit-regular if Mis smooth in a forwards-
backwards space-time neighborhood of any space-time point Xwith M(X) =1.
Note that we can then write sing M={X∈Rn+1×R:M(X) > 1}. Note that
by [46, Theorem 4.2] the class of unit-regular integral Brakke flows is closed under
the convergence of Brakke flows. Furthermore, combining [46, Lemma 4.1] and [51]
it follows that there is ε0>0, depending only on dimension, such that every point
X∈singMhas M(X) ≥1+ε0. Upper semi-continuity of density then implies
that singMis closed.
2.5 Cones and self-expanders
Consider S⊂Sna smooth, embedded, closed hypersurface. We then call the cone
over S, denoted by C=C(S) ⊂Rn+1,smooth. We say that M⊂Rn+1is a smooth
hypersurface with a conical singularity at x0modelled on the cone Cif:
(1) M\{x0}is a smooth (embedded) hypersurface,
(2) limρ→∞ ρ·(M −x0)=C,
where the convergence is in C∞
loc(Rn+1\{0}). Note that a hypersurface with conical
singularities is admissible and smoothable in the sense of Sect. 2.3, see also [22,
Appendix E].
Similarly, we say that a hypersurface M⊂Rn+1is (smoothly) asymptotic to Cif
lim
ρ0ρ·M=C
in C∞
loc(Rn+1\{0}).
A natural class of solutions to mean curvature flow, starting from an initial
(smooth) cone C,areself-similarly expanding solutions, i.e. solutions given by
t→√t·(2.2)
for t>0, where is asymptotic to C. These solutions are invariant under parabolic
rescalings forward in time. The condition that (2.2) is a mean curvature flow yields
an elliptic equation for , given by
H(x)−x⊥
2=0.(2.3)
We call aself-expander and denote the corresponding immortal solution to mean
curvature flow by M. Alternatively, self-expanders are critical points (under com-
pact perturbations) of the expander functional
E(M) =M
e|x|2
4dHn.
We call a self-expander stable if the second variation of Eis non-negative under
compact perturbations, i.e. if

ϕ(−Lϕ)e |x|2
4dHn≥0
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MCF from conical singularities 1051
for all ϕ∈C∞
c(), where Lis the corresponding Jacobi operator given by
L=+x
2·∇−1
2+|A|2.(2.4)
Note that a stable expander becomes strictly stable when restricting to any compact
subset K⊂. Denoting R:= ∩BR(0)for R∈(0,∞), this implies that there
exists a positive first eigenfunction φR∈C∞(R)(unique up to scaling) solving
LφR+μRφR=0inR
φR=0on∂R,(2.5)
where R→ μR>0 is monotonically decreasing in R. We will scale such that
Rφ2
Re|x|2/4=1, ensuring that φRis unique.
Linearising the expander equation (2.3) yields solutions to the linearized equation,
i.e. functions u∈C∞() such that Lu=0. We call such a function a Jacobi field.
We further recall the following decay estimate.
Proposition 2.3 ([31, Lemma 5.3]) Let denote an expander asymptotic to a smooth
cone.Then there is R>0sufficiently large so that \BR(0)can be written as a
normal graph over Cwith the graphical height function σ=o(|x|−1)as x→∞.
This improves the trivial σ=o(|x|)estimate via comparison with large spheres.
2.6 The level set flow of a cone and the outermost expanders
For a smooth cone C=C(S) with C=∂W for Wa closed set, we define Gap(C)to
be the level set flow of the cone Cat time t=1. Since the level set flow is unique,
and Cis invariant under scaling, it follows that the level set flow of Cis given by
t→√t·Gap(C)for t∈(0,∞).
The analogous statement to Proposition 2.2 holds also for the level-set flow of
smooth hypercones, see [22, Theorem E.2]. Furthermore, in [22, Theorem 8.21] it
was shown that the outermost/innermost flows from a cone (in low dimensions) are
modelled on smooth expanders, minizing the expander functional Efrom the outside.
(For n=2 smoothness had been shown by Ilmanen [38].) We will refer to these as
the outermost expanders. We summarize these facts as follows:
Theorem 2.4 ([22,38]) Fo r 2≤n≤6, let Cn⊂Rn+1be a smooth cone.Then,there
are smooth expanders ,,smoothly asymptotic to C.The expanders ,de-
scribe the level set flow of Cin the following sense:
•If the level set flow of Cdoes not fatten,then Gap(C)==:=.
•If the level set flow of Cdoes fatten,then ∩=∅and Gap(C)is the region
between and ,i.e.∂Gap(C)=∪.
Finally,minimizes the expander functional Eto the outside (relative to W)on
compact sets.Similarly,minimizes Eto the inside.
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1052 O. Chodosh et al.
Note that the property that ,minimize Efrom one side implies that both ,
are stable expanders. Furthermore, for n≥7, ,could apriorihave singular
set of dimension n−7. We say that the outermost flows of Care smooth if the singular
set is empty (so this always holds for 2 ≤n≤6). When the outermost flows of Care
aprioriknown to be smooth the proof of Theorem 2.4 carries over to prove the
remaining assertions in the theorem.
Let =W∩Sn. Recall that C=C(S) where S=∂ ⊂Snis a smooth embedded
hypersurface. Let νSbe the smooth unit normal vectorfield to Sin Snpointing to the
outside of .Givenψ:S→Ra positive, smooth function there exist ε>0 and a
smooth local foliation of hypersurfaces (Ss)−ε<s<ε in Snsuch that S0=Sand
∂
∂s Sss=0=ψ·νS.
We consider the cones Cs:= C(Ss)and the corresponding outermost expanders s,

s. Note that by construction of the outermost flows of Csit follows that for s>t
the outermost expander slies strictly to the outside (with respect to W)oftand
s→tsmoothly for st. Similarly for s<t the innermost expander 
tlies
strictly to the inside of 
sand 
t→
ssmoothly for st.
We denote with πthe composition of the closest point projection onto C(S) com-
posed with the radial projection C(S) →Sof the cone onto its link. This is well
defined on the cone over a neighborhood of Sin Sn. The next lemma then follows
from the above discussion together with [29, Lemma 2.2] and the strong maximum
principle.
Lemma 2.5 Let ψ:S→Rbe a positive,smooth function.Then there is a positive
Jacobi field von that satisfies
|∇ℓ
v|=O(r1−ℓ)
for ℓ=0,1,2,...,where r=|x|.Furthermore,the refined estimate
v=r·ψ◦π+w
with
|∇ℓ
w|=O(r−1−ℓ)
for ℓ=0,1holds.An analogous Jacobi field vexists on with the same asymptotic
expansions.
2.7 Forward rescaled flow
Given a (smooth) mean curvature flow (0,T) t→ M(t ) in Rn+1one obtains a
solution to forward rescaled mean curvature flow by considering the rescaling
(−∞,log(T )) τ→ ˜
M(τ) := e−τ/2M(eτ),
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MCF from conical singularities 1053
which satisfies the evolution equation
∂x
∂τ ⊥=H˜
M(τ)(x)−x⊥
2.
Note that expanders are the stationary points of this evolution.
3 Expander barriers
Let C=∂W denote a smooth cone so that the outermost flows of Care modeled on
smooth expanders ,. Assume that the level set flow of Cfattens (so and 
are distinct). Recall that the level set flow of Cis given by {√tGap()}t∈(0,∞)and
∂Gap() =∪.Below,weworkwithbut identical analysis for follows by
replacing Wwith W=Wc.
We choose the unit normal νpointing into Gap().
By Lemma 2.5,admits a positive Jacobi field of the form v=r+wwith
|∇ℓw|=O(r−1−ℓ). We also recall the definition of the first eigenfunction φRin
(2.5). For R>0 large and for α>0 to be fixed sufficiently small, we define
f3R,α =v+αφ3R(3.1)
on 3R:=∩B3R(0). Then, define
h=max
∂R
f3R,α >0.(3.2)
Then we define a function on all of by
u(x)=




f3R,α x∈R
min{f3R,α,h}x∈ER\2R
hx∈E2R,
(3.3)
where ER:= \BR(0). We want to check that for s>0 sufficiently small and α,
Rchosen appropriately, the (time-independent) family of hypersurfaces t→ s:=
graphsu is a supersolution to rescaled mean curvature flow (in the sense that a
rescaled mean curvature flow cannot touch sfrom below relative to its unit nor-
mal as fixed above). We start by checking that the graphs of hand f3R,α have good
intersection.
Lemma 3.1 There is R0=R0() so that for R≥R0there is α0=α0(R,  ) > 0
small so that if α∈(0,α
0)then h≥f3R,α on ∂Rand h≤f3R,α on ∂2R.
Proof The first inequality follows from (3.2). We now observe that (using the decay
of wobtained in Lemma 2.5)
h=max
∂R
f3R,α
≤R+O(R−1)+αmax
∂R
φ3R
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1054 O. Chodosh et al.
≤min
∂2R
f3R,α −R+O(R−1)+αmax
∂R
φ3R−min
∂2R
φ3R.
Taking Rsufficiently large so that the second and third terms satisfy −R+O(R−1)≤
−1 we can then take αsufficiently small (depending on Rthrough the dependence
of φ3R) so the fourth term is ≤1. This completes the proof. 
Thus it suffices to check that sh and sf3R,α define supersolutions on the appropri-
ate overlapping regions (for s>0 small).
Lemma 3.2 We can take R=R1() sufficiently large h0=h0() > 0sufficiently
small so that for h∈(0,h
0),τ→ R,h := graphERhdefines a supersolution to
rescaled mean curvature flow.
Proof Since is asymptotically conical, |A|<1onERfor Rsufficiently large.
Thus, R,h is smooth for R>0 sufficiently large. We compute
vR,h(x,τ)∂τxR,h −H+1
2xR,h·νR,h =−Lh+E(h)
=1
2−|A|2h+E(h)
≥1
2−|A|2h−C1h2
where we used Proposition A.3. Taking Rsufficiently large so that |A|2≤1
4. Then,
taking h0=1
4C1completes the proof. 
We now fix R=R0>0 sufficiently large so that both Lemmas 3.1 and 3.2 hold.
Then take α0as in Lemma 3.1.
Lemma 3.3 For an y α∈(0,α
0)there is s0=s0(, α) > 0sufficiently small,
τ→R0,α,s := graph2Rsf3R,α
is a supersolution to rescaled mean curvature flow for any s∈(0,s
0).
Proof Since 2Ris compact, R,α,s will be smooth as long as sis sufficiently small.
Moreover, we have
vR,α,s (x,τ)∂τxR,α,s −H+1
2xR,s·νR,s =−sLf3R,α +E(sf3R,α)
=sαμ3Rφ3R+E(sf3R,α).
Now we observe that since φ3R>0on3R, it holds that inf2Rφ3R>0. Combined
with μ3R>0 and the simple error estimate E(sf3R,α)=O(s2)(cf. Proposition A.3),
the assertion follows. 
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MCF from conical singularities 1055
We now fix α∈(0,α
0)and s0=s0(, α ) > 0 as in Lemma 3.3. Combining Lem-
mas 3.1,3.2, and 3.3 we obtain (recalling the definition of uin (3.3))
Proposition 3.4 There is s0>0sufficiently small so that τ→ s:= graphsu is a
supersolution to rescaled mean curvature flow for any s∈(0,s
0).
4 Fattening
We consider M⊂Rn+1with an isolated singularity at 0modeled on a smooth cone
C.LetUbe the compact region bounded by Mand write U=Uc. Define W,W⊂
Rn+1closed so that ∂W =∂W=Cand limρ→∞ ρU =W, limρ→∞ ρU=Win the
sense of local Hausdorff convergence. We let =∂F1(W) denote the outer expander
and =∂F1(W )the inner expander.
Approximate Mto the inside and outside by smooth hypersurfaces M
i⊂U,
Mi⊂Usatisfying conditions (1)-(5) described in Sect. 2.3.LetMi,M
idenote
unit-regular cyclic Brakke flows with Mi(0)=HnMi,M
i(0)=HnM
iobtained
via elliptic regularization. Passing to a subsequence we can assume that MiM,
M
iMunit regular cyclic Brakke flows with M(0)=M(0)=HnM.
Theorem 4.1 The flow Mis modeled on the expander near (0,0)in the sense that
limλ→∞ Dλ(M)=M.The same holds for M,namely limλ→∞ Dλ(M)=M.
In particular, this shows that if Cfattens under the level-set flow then so does M.
Before proving Theorem 4.1 we observe the following consequence. Recall that
M(t) (resp. M(t)) is the outer (resp. inner) flow of Mas defined in Proposition 2.2.
Corollary 4.2 There is T>0so that M{t<T}and M{t<T}are smooth and for
0≤t<T,we have supp M(t ) =M(t) (resp. supp M(t) =M(t)). Furthermore,
any unit regular integral Brakke flow ˇ
Mwith ˇ
M(0)=HnMand supp ˇ
M(t) ⊂
M(t) satisfies ˇ
M{t<T}=M{t<T}.
Proof It suffices to consider Mand outer flows. Suppose there are points Xi=
(xi,t
i)∈singMwith 0 <t
i→0. Since Mis smooth away from 0, it must hold that
xi→0. Suppose that up to a subsequence suppi|xi|2t−1
i<∞. Then by Theorem
4.1,Dt−1/2
i
(M)Mas i→∞. Since Dt−1/2
i
(Xi)=(xit−1/2
i,1)is bounded (by
our assumption), this contradicts the fact that is smooth. Thus, it remains to con-
sider the case that |xi|2t−1
i→∞. By Theorem 4.1 again, D|xi|−1(M)M.Up
to a subsequence, D|xi|(Xi)=(|xi|−1xi,|xi|−2ti)converges to (˜
x,0)with |˜
x|=1.
Since singM=(0,0), this is a contradiction. This proves the smoothness part of
the assertion.
By the work of Hershkovits–White as recalled in Proposition 2.2, the support of
Magrees with the outer flow of M. The final statement follows since t→M(t) is a
smooth mean curvature flow for 0 <t<T, so the constancy theorem implies that the
multiplicity of ˇ
Mis a non-negative constant for a.e. 0 <t <T, which additionally is
monotone in time. But the monotonicity formula together with unit regularity implies
that the multiplicity is one away from (0,0),so ˇ
Magrees with M.
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1056 O. Chodosh et al.
Proof of Theorem 4.1 It suffices to consider the outer flow M.Fixs∈(0,s
0)and
let sbe defined as in Proposition 3.4 with respect to . Recall that t→ √tsis
a supersolution to mean curvature flow and that the unit normal to points into
Gap(C).
Using the notation established in the beginning of this section, we have:
Claim 4.1 For i(j) →∞and λj→∞assume that ˜
Mj:=Dλj(Mi(j)) ˜
M. Then
supp ˜
M(t) ⊂W(t) (the level set flow of W).
Proof By construction, supp ˜
M(0)⊂W. Thus, the assertion follows from the avoid-
ance principle for Brakke flows (cf. [37, Theorem 10.7]). 
Claim 4.2 There is ρ0=ρ0(s) < ∞so that W(1)\Bρ0(0)lies strictly below s.
Proof Recall that W(1)is the region below . Thus, the assertion follows from the
fact that decays towards (cf. Proposition 2.3) whereas the function uin (3.3)is
constant near infinity. 
Claim 4.3 Fix ρ≥ρ0. There’s T=T(s,ρ) so that supp Mi(t) ∩B√tρ(0)lies below
√tsfor all t∈(0,T]and i∈N.
Proof of Claim 4.3 We first show that there T=T(s,ρ)>0 such that the claim holds
in a neighbourhood of ∂B√tρ(0). Assume this fails at times 0 <t
j→0forthe
flows Mi(j). Note that i(j) →∞as j→∞since Miis disjoint from M.Let
˜
Mj:= Dt−1/2
j
(Mi(j)). Pass to a subsequence so that ˜
Mj˜
M. By Claim 4.1,
supp ˜
M(t) ⊂W(t). However, by Claim 4.2,W(t ) \B√tρ0(0)lies strictly below
√tsfor t>0. Combined with upper semi-continuity of Gaussian density along
Brakke flow convergence (implying that points in the support of a converging se-
quence of flows converge to points in support) we obtain a contradiction.
Now, we note that suppMi(t ) is disjoint from B√tρ for all 0 ≤t≤2t(i) (since
Miis disjoint from 0). The claim then follows by applying the maximum principle in
B√tρ(0)from t(i) to T(s,ρ).
Letting i→∞establishes the same statement as in Claim 4.3 for with Mire-
placed by M. Since the flow t→ √tsis scaling invariant this implies that any
forwards blow-up of Mat (0,0)has to lie (weakly) below √ts∩B√tρ(0)for all
t>0. Letting ρ→∞and s→0 establishes that any forward blow-up4of ˜
Mis
supported (weakly) below t→ √t, starting at C. Since is the outer expander,
the support of ˜
Mthus must be a subset of t→ √t. Again, the constancy theorem
implies that the multiplicity is a non-negative constant for a.e. t>0, which addition-
ally is monotone in time. But the monotonicity formula together with unit regularity
implies that the multiplicity is one sufficiently far out, so any blow-up of Magrees
with t→√t.
4i.e. any subsequential limit of Dλ(˜
M)as λ→∞.
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MCF from conical singularities 1057
5 Global barriers
Let Cdenote a smooth cone over its link S⊂Snand a smooth, stable expander
asymptotic to C. We choose a global unit normal vector field on ν. As in Lemma
2.5 for the outermost expanders, we assume that admits a positive Jacobi field of
the form v=r·ψ◦π+w, where ψis a positive function on , together with the
decay |∇ℓw|=O(r−1−ℓ)for l=0,1.
We mimic part of the construction in Sect. 3. Since is stable, for every R>0
large we have a positive first eigenfunction φ3Rin (2.5) with eigenvalue μ3R>0. For
α>0 to be fixed sufficiently small, we take
f3R,α =v+αφ3R(5.1)
on 3R:=∩B3R(0). Then, define
h=max
∂R
f3R,α >0.(5.2)
We now consider M⊂Rn+1with an isolated singularity at 0modelled on C. Assume
that Mis a unit-regular cyclic Brakke flow, smooth on (0,T) for some T>0 with
M(0)=HnMand such that t−1/2M(t) converges smoothly to (on compact
subsets of Rn+1)ast0. Thus for every δ>0 and every R<∞there is Tδ,R and
a smooth family of functions u:3R×(0,T
δ,R)such that for all t∈(0,T
δ,R)
{x+u(x,t)ν
(x)|x∈3R}⊂t−1/2M(t) (5.3)
with u(·,t)C3(3R)≤δ
2.Fors∈(0,1)we define the close barriers
±
close,s (t) =√t·{(x+(u(x,t)±sf3R,α)·ν(x)) |x∈2R}
and the far barriers
±
far,s (t) ={(x±sh√t·νM(t)(x)) |x∈M(t) \B√tR(0)},
where we choose the unit normal vectorfield νM(t) such that it induces the same
orientations as νin the convergence t−1/2M(t) →as t0.
We aim to check that for s>0 sufficiently small and α,Rchosen appropri-
ately t→+
close/far,s (t) constitute supersolutions to mean curvature flow (in the sense
that a mean curvature flow cannot touch t→ +
close/far,s (t) from below relative to
its unit normal as fixed above) away from their respective boundaries. Similarly,
t→ −
close/far,s (t) constitute subsolutions to mean curvature flow (in the sense that
a mean curvature flow cannot touch t→−
close/far,s (t) from above relative to its unit
normal as fixed above) away from their respective boundaries.
To construct global barriers, we start by checking that ±
close,s and ±
far,s have good
intersection. This will be used to “weld” them together in the sense of Meeks–Yau
[42] to form a global barrier.
Lemma 5.1 There is R0=R0() so that for R≥R0there is α0=α0(R,  ) > 0
small and δ0=δ0(R,,α
0)>0and so that if α∈(0,α
0)and δ∈(0,δ
0)then for
t∈(0,T
δ,R)we have that
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1058 O. Chodosh et al.
(i) +
close,s (t) lies above +
far,s (t) in a neighborhood of ∂+
close,s (t).
(ii) +
far,s (t) lies above +
close,s (t) in a neighborhood of ∂+
far,s (t).
(i) −
close,s (t) lies below −
far,s (t) in a neighborhood of ∂−
close,s (t).
(ii) −
far,s (t) lies above −
close,s (t) in a neighborhood of ∂−
far,s (t).
for all s∈(0,1).
Proof This follows from Lemma 3.1 by choosing δ0>0 sufficiently small. 
Lemma 5.2 For R≥R0(),α∈(0,α
0),there is δ1=δ1(R,α,) > 0and s0=
s0(R,  ) > 0, such that for all s∈(0,s
0)and δ∈(0,δ
1),for t∈(0,T
δ,R),
t→+
close,s (t)
is a supersolution to mean curvature flow (and similarly t→−
close,s (t) is a subsolu-
tion to mean curvature flow).
Proof By rescaling it is equivalent to show that
τ→ ˜
(τ) := e−τ/2+
close,s (ln(τ ))
for τ∈(−∞,exp(Tδ,r)) is a supersolution to rescaled mean curvature flow. Similarly
we denote the corresponding solution of Mto rescaled mean curavture flow by
τ→ ˜
M(τ ) := e−τ/2M(ln(τ)) ,
which by (5.3) (writing ˜u(x,τ):=u(x,ln τ)) satisfies
{x+˜u(x,τ)ν
(x)|x∈3R}⊂ ˜
M(τ ) .
Applying Lemma A.4 we compute
v˜
∂τx˜
−H˜
+1
2x˜
·ν˜
=−sLf3R,α +Esf3R,α =αsμ3Rφ3R+Esf3R,α (5.4)
and provided ˜uC3(2R)+sf3R,αC3(2R)≤δwe can estimate pointwise
|Esf3R,α (x,τ)|≤Csδ(|f3R,α(x)|+|∇f3R,α(x)|+|∇2
f3R,α(x)|).
Note that φ3R>0on3R, so there exists C>0 such that for x∈2R
|f3R,α(x)|+|∇f3R,α(x)|+|∇2
f3R,α(x)|≤Cφ3R(x).
Thus (5.4) yields (for all s∈(0,s
0))
v˜
∂τx˜
−H˜
+1
2x˜
·ν˜
≥s(αμ3R−Cδ)φ3R>0,
on 2R, as long as δis sufficiently small. This yields the statement for t→+
close,s (t).
The statement for t→−
close,s (t) follows analogously. 
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MCF from conical singularities 1059
Take R0as in Lemma 5.1.
Lemma 5.3 There is R=R1(M,  ) ≥R0sufficiently large so that for R≥R1,taking
α0as in Lemma 5.1,s0as in Lemma 5.2 and Tfar >0sufficiently small depending on
M,,Rthe following holds:for t∈(0,T
far)the flow t→ +
far,s (t) is a supersolution
to mean curvature flow,away from its boundary (and similarly t→ −
far,s (t) is a
subsolution to mean curvature flow,away from its boundary).
Proof We first establish the following claim. Consider the n-dimensional ball in the
{xn+1=0}hyperplane, given by Bn
r(0):=Br(0)∩{xn+1=0}.
Claim 5.1 There exists η=η(n) > 0 with the following property. Let u:Bn
1(0)×
[0,1]with u(·,t)C3(Bn
1(0)) ≤ηsuch that for t∈[0,1]
M(t) :={(ˆ
x,u(ˆ
x,t))|ˆ
x∈Bn
1(0)}
constitutes a smooth mean curvature flow. Let νM(·,t) be the upwards unit normal
to M(t). Consider
±(t) :={x±s√tνM(x,t)|x∈M(t)}.
Then for s∈(0,1), the flow (0,1]t→+(t ) is a supersolution to mean curvature
flow. (Similarly, the flow (0,1]t→−(t ) is a subsolution to mean curvature flow.)
Proof of Claim We consider +(t) (the computation for −(t ) is analogous). Let t→
xM(t) be a point evolving normally, i.e. ∂txM=HM(xM). Recall that
|∂tνM(xM(t ), t)|=|∇MHM|≤Cη.
Thus if we set x+=xM+s√tνMthen
∂tx+=HM+(∂t(s√t ))νM+s√t(∂tνM)
so this point is evolving with normal speed
v+(∂tx+)·ν+=HM+∂t(s√t) +s√t(∂tνM)·ν+
where v+=(ν+(x+)·νM(xM))−1(cf. Appendix A.3). Thus
v+(∂tx+−H+)·ν+=HM−v+H++∂t(s√t) +s√t(∂tνM)·ν+
=∂t(s√t) −|AM|2s√t+E,
where the error satisfies |E|≤Csη√t, where Cdepends only on n(using Lemma
A.2 to estimate the difference in mean curvatures and the computation above to esti-
mate the unit normal evolution term). Thus, for η>0 sufficiently small and t∈(0,1]
v+(∂τx+−H+)·ν+≥(1
2t−Cη)s√t≥(1
2−Cη)s√t>0.
This establishes the claim. 
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1060 O. Chodosh et al.
There is CM>0 so that |y||AM(y)|+|y|2|∇AM(y)|≤CM(this follows because
Mis modeled on a smooth cone at 0). This implies that after choosing Rsufficiently
large, we have for all x∈M\{0}that
(R|x|−1(M −x)) ∩B2(0)
can be written as a C3-graph over its tangent plane with C3-norm bounded by
O(R−1). By pseudolocality (cf. [41, Theorem 1.5] or [18, Theorem 7.3]) this im-
plies that (taking Rsufficiently large)
(R|x|−1(M(R−2|x|2t)−x)) ∩B1(0)
remains a small Lipschitz graph over its tangent plane for t∈[0,2]. Parabolic esti-
mates then imply that if we take Reven larger, the graph will have C3-norm bounded
by η(as defined in Claim 5.1).
We can thus apply the scaled version of Claim 5.1 over balls at x∈Mof radius
R−1|x|. The claim only applies for times t∈[0,R−2|x|2], but for larger times, the
graph in M(t) over this ball will be contained in B√tR, and thus does not contribute
to +
far,s (t). This completes the proof. 
We now choose R≥R1to satisfy Lemma 5.3. We then choose α∈(0,α
0)as in
Lemma 5.1 and then s0as in Lemma 5.2.
Proposition 5.4 There is δ>0sufficiently small such that for all s∈(0,s
0)and
t∈(0,T
δ,R),we can weld t→ +
close,s (t) to t→ +
far,s (t) to obtain a global super-
solution t→+
s(t) to mean curvature flow.Similarly,we can weld t→−
close,s (t) to
t→−
far,s (t) to obtain a global subsolution t→ −
s(t) to mean curvature flow.
6 Uniqueness
We work with the same set-up as in the previous section.
Assume that we have smooth mean curvature flows M1and M2, defined on
(0,T) for some T>0, with M1(0)=M2(0)=HnMand such that t−1/2Mi(t)
converges smoothly to (on compact subsets of Rn+1)ast0fori=1,2.
Lemma 6.1 It holds that dist(M1(t ), M2(t)) =o(√t).
Proof Assume that there is κ>0 and xi∈suppM1(ti)with ti→0 so that
d(xi,suppM2(ti)) ≥κ√ti.(6.1)
Pass to a subsequence so that xi→x∈M.Ifx=0then (ti)−1/2Mi(ti)both con-
verge to in C∞
loc(Rn+1).Ifx= 0then short-time smoothness of Mi(t ) around x
gives that5(ti)−1/2(Mi(ti)−x)both converge to TxMin C∞
loc(Rn+1). In either case,
we see that assuption (6.1) cannot hold. This completes the proof. 
5Note that one can view the x=0case as the “same” as the x=0case, since Mis modeled on the cone
TxMat x=0, which evolves as a static hyperplane under level set flow.
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MCF from conical singularities 1061
Proposition 6.2 It holds that M1(t) =M2(t) on [0,T).
Proof By Proposition 5.4 we can construct for all s∈(0,1)and t∈(0,T
δ,R)the
supersolution t→ +
s(t) and subsolution t→ −
s(t) over M1. Note that for any
s∈(0,s
0),wehavebyLemma6.1 that M2(t) lies between +
s(t) and −
s(t) for
t>0 sufficiently small. This yields for all s∈(0,s
0)and t∈(0,T
δ,R)that M2(t) lies
between +
s(t) and −
s(t). Since for all t∈(0,T
δ,R)we have that +
s(t ), −
s(t) →
M1(t) as s→0 this yields that M1coincides with M2on (0,T
δ,R). Thus M1and
M2have to coincide (at least as long as they both remain smooth). 
Corollary 6.3 If Cdoes not fatten under the level set flow,then the inner and outer
flows agree for t∈[0,T].
Proof If Cdoes not fatten then =. Thus, by Theorem 4.1, the outer and inner
flows M,Mare both modeled on near (0,0). The assertion then follows from
Proposition 6.2.
Appendix: Graphs over expanders
We consider a smooth embedded hypersurface M⊂Rn+1and assume that νMis a
choice of smooth unit normal vector field to M. We define its shape operator (or
Weingarten map)
Sp:TpM→TpM, ξ →−Dξν(A.1)
and its second fundamental form
A:TpM×TpM→R,(ξ,ζ)→ A(ξ , ζ ) =Sp(ξ) ·ζ. (A.2)
We fix the sign of the scalar mean curvature Has follows
H=Hν
,
and thus H=tr S=tr A, with the principal curvatures of Mbeing the eigenvalues of
S. We consider u:M×I→Rso that
|u||AM|<η<1
along M×I, where Ais the second fundamental form of . This allows us to define
the graph
τ:={x+u(x,τ)ν
M(x):x∈M}.
We compute here various geometric quantities associated to τ. The computations
follow directly as in [21, Appendix A]. There the focus was on the backwards rescaled
flow, but the computations for the forwards rescaled flow are completely analogous
and just amount to changing sign in front of the corresponding terms.
Define
v(x,τ)=(1+|(Id −uSM)−1(∇Mu)|2)1
2.(A.3)
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1062 O. Chodosh et al.
Lemma A.1 ([21, Lemma A.1]) The upwards pointing normal along τis
ν=v−1(−(Id −uSM)−1∇Mu+νM). (A.4)
In particular
v=(ν·νM)−1.(A.5)
For ℓ∈{0,1}set
σℓ(x,τ):=
ℓ

j=0
4−2j

k=0|∂j
τ∇ku(x,τ)|.(A.6)
Lemma A.2 ([21, Lemma A.2]) The mean curvature of τat x+u(x,τ)ν
M(x)satisfies
v(x,τ)H
(x+u(x,τ)ν
M(x), τ ) =HM(x)+(Mu+|AM|2u)(x)+EH(A.7)
where the error EHcan be decomposed into terms of the form
EH=uEH
1+EH
2(∇Mu, ∇Mu)
where EH
1∈C∞() and EH
2∈C∞(;T∗M⊗T∗M) satisfy the following esti-
mates:
|∂τEH
1(x,τ)|≤CH
1σ1(x,τ),
2

k=0|∇k
EH
1(x,τ)|≤CH
1σ0(x,τ)
and6
|∂τEH
2(x,t)|≤CH
2(1+σ1(x, τ )),
2

k=0|∇kEH
2(x,t)|≤CH
2(1+σ0(x,τ))
where CH
1,CH
2depend only on ηand an upper bound for 3
k=0|∇kA|(x).
Observe that the expander mean curvature can be written as
H−x⊥
2=H−1
2x·νν
and recall the definition of the (expander) Jacobi operator L,see(2.4). For the fol-
lowing we assume that M=, where is an (open subset of an) expander.
Proposition A.3 ([21, Corollary A.4]) We have
v(x, τ )(∂τx−H+1
2x)·ν=∂τu−u +1
2xT·∇u+(|A|2−1
2)u
 
=Lu +E
6recall that E2is a section of T∗⊗T∗so e.g., ∇E2is a section of T∗⊗T∗⊗T∗.
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MCF from conical singularities 1063
at x=x+u(x,τ)ν
(x)for E=uE1+E2(∇u, ∇u) for E1,E2satisfying
|∂τE1(x,τ)|≤C1σ1(x,τ),
2

k=0|∇kE1(x,τ)|≤C1σ0(x,τ)
and
|∂τE2(x,τ)|≤C2(1+σ1(x, τ )),
2

k=0|∇kE2(x,τ)|≤C2(1+σ0(x,τ))
where C1,C2depend only on ηand an upper bound for 3
k=0|∇k
A|(x).
In particular,if τis a solution to rescaled mean curvature flow,i.e.,
(∂τx)⊥=H−1
2x⊥
then
∂τu=Lu +E(A.8)
for Eas above.
We also need to understand the linearization of the expander mean curvature of
two graphs over , relative to each other and relative to the base .
Lemma A.4 ([21, Lemma A.7]) For δ<1
2(sup|A|)−1and ui∈C∞(),i=0,1
with
|ui(x)|+|∇ui(x)|+|∇2
ui(x)|+|∇3
ui(x)|≤δ(A.9)
for all x∈.Letting
i:={x+ui(x)ν(x):x∈}
then denoting w=u1−u0and vi:= (νi·ν)−1there exists C=C(sup|A|+
|∇A|+|∇2A|)such that
v1H1−1
2x1·ν1−v0H0−1
2x0·ν0=Lw+Ew
where xi=x+ui(x)ν(x)and the error term Esatisfies
Ew(x)=w(x)F (x)+∇w(x)·F(x)+∇2w(x)·F(x),
with the estimate
|F|+|F|+|F|+|∇F|≤Cδ
for all x∈.
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1064 O. Chodosh et al.
Funding O.C. was supported by a Terman Fellowship and an NSF grant (DMS-2304432). J.M.D-H. was
supported by the Warwick Mathematics Institute Centre for Doctoral Training. This project has received
funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research
and innovation programme, grant agreement No 101116390.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
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4.0/.
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