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arXiv:2312.10940v1 [math.DG] 18 Dec 2023
RIGIDITY OF AREA NON-INCREASING MAPS
MAN-CHUN LEE1, LUEN-FAI TAM, AND JINGBO WAN
Abstract. In this work, we consider the area non-increasing map between
manifolds with positive curvature. By exploring the strong maximum prin-
ciple along the graphical mean curvature flow, we show that an area non-
increasing map between positively curved manifolds is either homotopy triv-
ial, Riemannian submersion, local isometry or isometric immersion. This
confirm the speculation of Tsai-Tsui-Wang. We also use Brendle’s sphere
Theorem and mean curvature flow coupled with Ricci flow to establish re-
lated results on manifolds with positive 1-isotropic curvature.
1. introduction
People have been interested to study properties of maps between compact
Riemannian manifolds in terms of the so-called k-dilation. The 1-dilation maps
(with constant 1), which are just distance non-increasing maps, have been
studied intensively, see [9, 10, 13, 15] for example. The 2-dilation maps (with
constant 1) are maps which are (two dimensional) area non-increasing. In case
the constant is less than 1, then it will be called (strictly) area decreasing map
in this work. Interesting and important results have also been obtained. In
particular using graphical mean curvature flow, Tsui-Wang [25] confirmed a
conjecture of Gromov [11] that any map from the standard sphere Sminto Sn
with 2-dilation constant less than 1 must be homotopically trivial. Later Lee-
Lee [16] proved that under certain curvature conditions, any area decreasing
map between compact manifold is also homotopically trivial, see also [2, 22,
23, 17]. Recently, more general results are obtained by Tsai-Tsui-Wang [26]
in this direction. It is well-known that in contrast, by the work of Guth [12]
one cannot expect an analogous result for k≥3. Motivated by the work
of Tsai-Tsui-Wang [26], in the first part of this work we want to study area
non-increasing map under similar curvature conditions in their work. For area
non-decreasing map, there is a well-known rigidity result by Llarull [21] which
says that any area non-increasing map from a spin n-manifold (Nn, g) with
scalar curvature R(g)≥n(n−1) to the standard n-sphere with nonzero degree
must be an isometry. Since the results of Tsai-Tsui-Wang [26] do not assume
that the manifolds are spin, one may wonder if there are similar results for
Date: December 19, 2023.
2020 Mathematics Subject Classification. Primary 51F30, 53C24.
1Research partially supported by Hong Kong RGC grant (Early Career Scheme) of Hong
Kong No. 24304222, 14300623, a direct grant of CUHK and NSFC grant No. 12222122.
1
2 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
area non-increasing map in the setting of [26]. Motivated by this, in the first
part of this work, we prove the following:
Theorem 1.1. Let (Mm, g),(Nn, h)be two compact manifolds with m, n ≥
3. Suppose f0is a smooth map from Mto Nwhich is area non-increasing.
Suppose one of the following curvature conditions is satisfied:
(i)
Ricg
min −Rich
max + (m−ℓ)·κM+ (n−ℓ)·κN≥0,and κM+κN>0;
or
(ii)
κM>0and (ℓ−1)τN≤(2(m−ℓ) + ℓ−1) κM.
with ℓ= min{m, n}.
Here κM, κNare the lower bounds of the sectional curvature of M, N respec-
tively, τNis the upper bound of the sectional curvature of N,Ricg
min is the
minimum of the eigenvalues of Ricgin M, and Rich
max is the maximum of the
eigenvalues of Richin N.
Then either f0is homotopically trivial or f0is a Riemannian submersion
(if m > n), local isometry (if m=n) or isometric immersion (if m < n).
See Theorem 3.1 for more details. Among other things, as a corollary, one
can conclude that any area non-increasing self map from CPn,n≥2 with
standard Fubini-Study metric is either an isometry or is homotopically trivial.
This confirms a conjecture (‘speculating’ in their words) of Tsai-Tsui-Wang
[26]. In particular, any area non-increasing self map from CPnwith non-zero
degree must be an isometry. On the other hand any area non-increasing map
from Smto Snwith n > m ≥3 and with standard metrics must be either
homotopically trivial or a Riemannian immersion or both. We should remark
that the conditions (i), (ii) above are not totally unrelated. In fact, one can
check that (ℓ−1)τN≤(2(m−ℓ) + ℓ−1) κMwill imply Ricg
min −Rich
max +(m−
ℓ)·κM+ (n−ℓ)·κh≥0. Moreover, condition (i) will imply that Ricg≥0.
The second main theme of this work is to generalize the work of Tsui-Wang
[25] on spheres in a different direction. In [5], Brendle showed that compact
manifolds with positive 1-isotropic curvature are diffeomorphic to quotient
of sphere using the Ricci flow, which is a generalization of Brendle-Schoen’s
differentiable sphere theorem in [4]. Let us recall the definition of positive
1-isotropic curvature.
Definition 1.1. We say that a curvature type tensor Ris in the cone of
positive 1-isotropic curvature, i.e. R∈CP I C1, if for all orthonormal frame
{ei}4
i=1 and µ∈[0,1], we have
R1331 +µ2R1441 +R2332 +µ2R2442 −2µR1234 >0.
Rigidity of area non-increasing maps 3
For a Riemannian manifold (M, g), we define
(1.1) χIC 1(g) = sup s∈R: Rm(g)−s·1
2g?g∈CP IC 1
where ?denotes the Kulkarni-Nomizu product, see (2.11). We say that (M, g)
has non-negative 1-isotropic curvature if χIC1(g)≥0.
One can equivalently interpret the non-negative 1-isotropic curvature using
the language of lie algebra by the work of Wilking [29]. In dimension three, the
latter definition is still well-defined and is equivalent to Ric ≥0. Therefore,
χIC 1(g)≥0 will be understood to be Ric ≥0 when n= 3.
Motivated by Brendle’s sphere Theorem, we are interested in the homotopy
problem on manifolds with positive 1-isotropic curvature. By considering the
graphical mean curvature flow coupled with the Ricci flows, we prove the
following theorem:
Theorem 1.2. Let (Mm, g)be a compact, locally irreducible and locally non-
symmetric manifold with χIC1(g)≥0and (Nn, h)be a compact manifold such
that m, n ≥3. Suppose f0is a smooth map from Mto Nwhich is area
non-increasing. Suppose
(1.2) Rmin(g)≥m
nRmax(h)
and one the following curvature conditions is satisfied
(i) (N, h)is Einstein with κN≥0; or
(ii) τN≤0.
Here Rmin(g)denotes the minimum of scalar curvature of gand Rmax(h)
denotes the maximum of scalar curvature of h. Then either f0is homotopically
trivial or f0local isometry (if m=n) or isometric immersion (if m < n).
See Theorem 3.3 for more details. If gis assumed to have χIC 1(g)>0,
i.e. positive 1-isotropic curvature, then gis locally irreducible. Then gmust
be either locally non-symmetric or a quotient of sphere by Brendle’s sphere
Theorem [5]. In either case, we might apply Theorem 1.1 or 1.2 to discuss the
area non-increasing maps ffrom Mto N.
The paper is organized as follows. In Section 2, we discuss the preliminary
on geometry of graphical mean curvature flow, its short-time and long-time
existence. In Section 3, we will prove the monotonicity along the graphical
mean curvature under various curvature conditions and will use them to prove
rigidity of area non-increasing map.
Acknowledgement: The authors would like to thank Prof. Chung-Jun Tsai,
Prof. Mao-Pei Tsui and Prof. Mu-Tao Wang for valuable discussions and
answering some of the questions. The first named author would like to thank
Prof. Jason Lotay for some insightful discussion on coupled flows. The third
named author would like to thank Prof. Mu-Tao Wang for his continuing
support, and also for introducing this problem.
4 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
2. Preliminaries: graphical mean curvature flow, short time
and long time existence
Let (Mm, g(t)), (Nn, h(t)) be two compact manifolds with smooth families of
metrics g(t), h(t) which may be independent of time. In this section, we discuss
the mean curvature flow from Mto Xm+n=Mm×Nnwith G(t) = g(t)⊕h(t)
is the product metric. We will concentrate on graphical solutions. The short
time existence is well-known [1, §6.4] and the ‘long time’ existence is now
standard because of the work of Wang [27]. However, we will sketch the
proofs on the existence the solution for the convenience of the readers. First
let us recall the mean curvature flow equation.
2.1. Mean curvature flow equation and short-time existence. Let us
first recall the setting of the mean curvature flow. Let (Mm, η(t)), (XQ, G(t))
be a compact manifolds where G(t) is a smooth family of metrics on X,t∈
[0, T ), T > 0. Let ∇,e
∇be Riemannian connections on (M , η(t)), (X, G(t)).
Let F:f
M=: M×[0, T )→Xbe a smooth map. Consider in local coordinates
{xi}in M,{yα}in X, a section of ⊗k(T∗(f
M))⊗F−1(T(X)) is of the following
form:
s=sα
i1...ikdxi1⊗ · · · ⊗ dxik⊗∂yα.
Then at (x, t):
s|p=D∂xps
=sα
i1...ik;pdxi1⊗ · · · ⊗ dxik⊗∂yα+sα
i1...ikdxi1⊗ · · · ⊗ dxik⊗e
∇F∗(∂xp)∂yα
=sα
i1...ik;pdxi1⊗ · · · ⊗ dxik⊗∂yα+Fβ
psα
i1...ikdxi1⊗ · · · ⊗ dxik⊗e
∇∂yβ∂yα
(2.1)
Here ; is the covariant derivative with respect to η(t). If there is no confusion,
we will also use ; to denote the covariant derivative with respect to G(t). Also,
s|t=D∂ts
=∂tsα
i1...ikdxi1⊗ · · · ⊗ dxik⊗∂yα+sα
i1...ikdxi1⊗ · · · ⊗ dxik⊗e
∇F∗(∂t)∂yα
=∂tsα
i1...ikdxi1⊗ · · · ⊗ dxik⊗∂yα+Fβ
tsα
i1...ikdxi1⊗ · · · ⊗ dxik⊗e
∇∂yβ∂yα
(2.2)
Suppose F0:M→Xbe an immersion. The mean curvature flow equation is
given by:
(2.3) ∂tF=H, on M×[0, T ).
F|t=0 =F0.
where Ft:M→Xwith Ft(x) = F(x, t) is an immersion, and H=H(t) is the
mean curvature vector of Ft(M) with respect to G(t). Let η(t) = F∗
t(G(t)).
We define ∆η(t)s=ηijs|ij then the mean curvature flow is of the form:
(2.4) ∂tF= ∆η(t)F.
Rigidity of area non-increasing maps 5
We have the following short time existence result.
Lemma 2.1. In the above setting, let F0:M→(X, G(0)) be a smooth im-
mersion. Then there is T0>0such that (2.3) has a solution on M×[0, T0).
Namely, Ftis an immersion, and satisfies (2.4) with η(t) = F∗
t(G(t)). More-
over, the solution is unique.
Proof. The proof is exactly as the case when G(t) is a fixed metric, see [1,
§6.4]. We sketch the proof. Let hbe a fixed metric on Mwith connection Γ.
Consider the parabolic systems in local coordinates
(2.5) ∂tFα=ηij Fα
ij −Γk
ijFα
k+ Γα
βγ Fβ
iFγ
j.
This is a global solution: it does not depend on local coordinates, as
∂tFα=ηij Fα
ij −Γk
ij Fα
k+ Γα
βγ Fβ
iFγ
j+ηij(Γk
ij −Γk
ij )Fα
k
Since this is strictly parabolic and Mis compact, it admits a short time
solution with initial map F0by standard parabolic theory. If tis small, then
Ft(p) = F(p, t) is also immersion. Let us assume this is the case. Consider the
vector field W(p, t) such that dFt(W) = ηij (Γk
ij −Γk
ij )Fα
k∂yαwhich is tangent
to Ft(M). Since Ftis an immersion, Wexists. Consider the solution of ODE:
(2.6) d
dt φ(p, t) = −W(φ(p, t), t)
φ(p, 0) = p.
Then p7→ φ(p, t) is a diffeomorphism on M. Let
e
F(p, t) = F(φ(p, t), t).
Then
∂te
F(p, t) =∂tF(φ(p, t), t) + dFt◦dφ(∂t)
=∂tF(φ(p, t), t)−dFt(W)
=∆η(t)F|φ(p,t)
=∆φ∗
t(η(t)) e
F|φ−1
t(φ(p,t))
=∆eη(t)e
F
where η(t) = F∗
tG(t) and eη(t) = e
F∗
tG(t). The uniqueness follows from the fact
that both solutions to (2.5) and (2.6) are unique provided the initial data are
the same, see [1, §6.4].
2.2. Geometry of graph and area non-increasing maps. We now spe-
cialize to the case when X=Mm×Nn,G(t) = g(t)⊕h(t) is an evolving
product metric and the initial data is a graph, i.e. F0= Id ×f0:M→X. Let
F:M×[0, T ]→Xbe the solution obtained from Lemma 2.1, it is easy to
see that the solution remains graphic for a short time. We assume working
6 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
on [0, T ] where Fremains graphical and will therefore call it to be a solu-
tion to the graphical mean curvature flow coupled with G(t). In what follows,
we will abbreviate it as graphical mean curvature flow when the context is
clear. By graphical condition, there exists smooth family of diffeomorphism
φt∈Diff(M) and maps ft:M→Nsuch that Ft= (Id ×ft)◦φtfor each
t∈[0, T ]. At each (p, t)∈M×[0, T ], we let λ2
1≥... ≥λ2
mbe the eigenvalues
of f∗
thwith respect to gat x=φt(p) so that the corresponding λi≥0 are the
singular values of dftat x. The map ft: (M, g(t)) →(N, h(t)) is said to be:
(2.7)
distance non-increasing, if λi≤1 for all i;
distance decreasing, if λi<1 for all i;
area non-increasing, if λiλj≤1 for all i6=j;
area decreasing, if λiλj<1 for all i6=j.
The meaning of the terminology is obvious. In order to emphasis, sometimes
we will call area decreasing map as strictly area decreasing map. Following
[25], to detect whether ftis area non-increasing, we introduce the following
tensors to detect whether ftis distance non-increasing etc. Let πM, πNthe
projections of M×Nonto M, N respectively. Let
(2.8) s(t) = π∗
Mg(t)−π∗
Nh(t),
and let S(t) = F∗
ts(t) be a 2-tensor on Mwhich in local coordinate is given
by
(2.9) Sij =Fα
iFβ
jsαβ.
If there is no possible confusion, we will also π∗
Mg(t) by g(t) for example.
Let Θ = S?η. In local coordinate,
(2.10) Θijkl =Silηjk +Sj kηil −Sikηjl −Sj lηik.
Here ?be the Kulkarni-Nomizu product: namely, for symmetric (0,2) tensors
S, T on a manifold,
(S?T)(X, Y, Z, W ) :=S(X, W )T(Y , Z) + S(Y, Z)T(X, W )
−S(X, Z)T(Y , W )−S(Y, W )T(X, Z)
(2.11)
for tangent vectors X, Y, Z, W . Equivalently one can write
Θ = F∗
tg?F∗
tg−F∗
th?F∗
th.
Furthermore, Θ is a curvature type tensor and can be considered as a sym-
metric bilinear form on Λ2T∗Mso that
Θ(X∧Y, Z ∧W) = Θ(X, Y, W, Z) = (S?η)(X, Y, W, Z).
Hence if e1, e2is an orthonormal pair, then
Θ(e1∧e2, e1∧e2) = Θ(e1, e2, e2, e1).
Rigidity of area non-increasing maps 7
In our convention, the sectional curvature of the two plane spanned by
orthonormal pair e1, e2is given by
R(e1, e2, e2, e1).
If gis a Riemannian metric, then
(g?g)(X, Y, Y, X) = 2 g(X, X)g(Y, Y )−(g(X, Y ))2
which is just twice of the area of the parallelogram spanned by X, Y . Hence
it is easy to see that S≥0 with respect the metric η(t) if and only if ftis
distance non-increasing, and Θ ≥0 as a bilinear form on Λ2T∗Mwith respect
the metric 1
2η?ηif and only if ftis area non-increasing. More precisely, we
have:
Lemma 2.2. Given a graphical embedding F:M→M×Nsuch that F=
(Id ×f)◦φfor some map f:M→Nand diffeomorphism φ∈Diff(M), then
(i) fis distance non-increasing (decreasing resp.) if and only if S≥0
(>0resp.) on M;
(ii) fis area non-increasing (decreasing resp.) if and only if Θ≥0(>0
resp.) on M.
We will also use the notation that Θ ≥ato mean that Θ −a
2η?η≥0 as a
symmetric bilinear form on Λ2T∗M.
To ease our computation, it is easier to choose a frame which is compatible
with the graphical structure. To do this, for p∈Mwe let {ui}n
i=1 be a g-
orthonormal frame at x=φt(p)∈Mand {vi}m
i=1 be a h-orthonormal frame
at ft(x)∈Nso that dft(ui) = λivifor all 1 ≤i≤min{m, n}=: ℓ. We extend
λito be 0 for i > min{n, m}for notation convenience. For metrics g(t) and
h(t), we will then use Kg
ij , Kh
ij to denote the sectional curvature of the plan
Σ = span{ui, uj}and Σ′= span{vi, vj}. Their curvature and evolution of
metrics are defined analogously.
With {ui}m
i=1 and {vi}n
i=1, we define
(2.12)
ei=ui+λivi
p1 + λ2
i
,for 1 ≤i≤m
νa=−λaua+va
p1 + λ2
a
,for 1 ≤a≤n
so that {ei}m
i=1 forms an orthonormal basis on T(x,ft(x)) Ft(M) while {νa}n
a=1
forms an orthonormal basis on N(x,ft(x)) Ft(M) with respect to the product
metric G. Thus, dπM(ei) = (1 + λ2
i)−1/2uiand hence the map ftwill stay
graphical if λiis bounded uniformly. To unify the notation, we will denote
this orthonormal frame as {˜
Eα}m+n
α=1 by defining ˜
Eα=eαfor 1 ≤α≤mand
˜
Eα=νafor α=m+awhere 1 ≤a≤n. With respect to this singular
8 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
decomposition, we then have ηij =δij and
(2.13)
Sij =1−λ2
i
1 + λ2
i
δij ;
Θijj i =Sii +Sjjfor i6=j;
Cii =: 2λi
1 + λ2
i
at the point (x0, t0)∈M×[0, T ] under consideration. Note that S2
ii +C2
ii = 1
for 1 ≤i≤m.
Since ei∧ejfor i < j are orthonormal frames of Λ2T∗Mwith respect to
the metric 1
2η?η, one can see that
{Θijj i|1≤i < j ≤m}
are indeed the eigenvalues of Θ with respect to 1
2η?η. Furthermore for
1≤i, j ≤mand 1 ≤a, b ≤n,
(2.14)
s(˜
Ei,˜
Ej) = s(ei, ej) = Sij ;
s(˜
Ei,˜
Em+a) = −2λi
1 + λ2
i
δia;
s(˜
Em+a,˜
Em+b) = λ2
a−1
1 + λ2
a
δab.
We abbreviate it as graphical frame. We will use it when applying tensor
maximum principle.
2.3. Long-time existence. Recall that a graphical mean curvature flow is
given by a graph of ft: (M, g(t)) →(N, h(t)) where one can use the graphic
frame (2.12). Let λibe the eigenvalues of f∗
t(h(t)) as in the definition of the
graph frame and let Θ be as in (2.13). Following the argument of Wang [27]
which is based on Huisken’s monotonicity [14], we prove the following long-
time existence criteria with possibly evolving background.
Theorem 2.1. Let (Mm, g(t)) and (Nn, h(t)) be two smooth compact evolving
families on [0, T )with T≤+∞. Denote G(t) = g(t)⊕h(t)be the product
metric on X=M×N. Suppose F:M×[0, t0)→(X, G(t)) is a smooth
solution of the mean curvature flow coupled with G(t)with t0< T . If for each
t,Ftis a graph given by strictly area decreasing map ft:M→Nso that
Θ(·, t)≥δfor some δ > 0on [0, t0), then the flow can be extended beyond t0
and remains graphical.
Proof. Since t0< T , both g(t) and h(t) has bounded geometry of infinity order
on [0, t0]. Moreover since Θ ≥δ > 0 on M×[0, t0), we have 0 ≤λi≤2
δby
(2.13) on M×[0, t0), see [26, Lemma 3.3] for example.
In what follows, we will use Cito denote any positive constant depending
only on G|[0,T0], δ, n, m, X. By the proof of [26, Theorem 3.2] (see Corollary
Rigidity of area non-increasing maps 9
3.1 below), and the fact that λi≤2
δ, we have
∂
∂t −∆ηlog det Θ
1
2η?η≥a|A|2−C2.
for some a > 0 depending only on m, n and |A|is the norm of the second
fundamental form. Here η(t) = F∗
t(G(t)). Since 0 ≥log det Θ
1
2η?ηand Θ ≥δ, we
can find C > 0 such so that φ=C−log det Θ
1
2η?ηsatisfies 1 ≤φ≤Cand
(2.15) ∂
∂t −∆ηφ≤ −a|A|2+C2.
We want to follow the argument of Wang [27]. We isometrically embed
(X, G(t0)) into RQ. Denote its image also by X. In a tabular neighborhood U
of X, extend G(t) smoothly to e
G(t) in Uso that e
G(t0) is the Euclidean metric
on U. This can be done since G(t) is smooth on X. Now we re-write the mean
curvature flow equation as
(2.16) ∂tF=H=e
H+E1
where E1=−trFt(M)A(U,
e
G(t));(X,G(t)) and e
His the mean curvature of Ft(M)
in (U, e
G(t)) and A(U,
e
G(t));(X,G(t)) is the second fundamental form of (X, G(t))
in (U, e
G(t)). We consider the (m-dimensional) backward heat kernel in RQ
centered at (y0, t0): for t < t0and y∈RQ,
(2.17) ρy0,t0(y, t) = 1
(4π(t0−t))m/2·exp −d2
euc(y, y0)
4(t0−t).
where deuc(x, y) = |x−y|denotes the standard Euclidean distance between
x, y ∈RQ. We will also use •to denote the Euclidean inner product and
denote τ(t) = t0−t. By translation, we assume y0to be the origin. Consider
the function ρ(x, t) = ρy0,t0(F(x, t), t) on M×[0, t0). By differentiating ρwith
respect to x, t, we have
(2.18)
∂tρ=ρm
2τ−|F|2
4τ2−∂tF•F
2τ;
ρi=−ρ·Fi•F
2τ;
ρ;ij =ρ·Fi•F
2τ
Fj•F
2τ−ρ·F;ij •F+Fi•Fj
2τ
where ; is the covariant derivatives with respect to η=F∗Gwhich is the
induced metric of G(t) on the submanifold Ft(M). Hence,
∂
∂t + ∆ηρ=ρm
2τ−ηij Fi•Fj
2τ+ηij(Fi•F)(Fj•F)
4τ2−|F|2
4τ2
−ρ·((∂t+ ∆η,euc)F)•F
2τ.
(2.19)
10 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
Since (X, G(t)) is isometrically embedded into (RQ,e
G(t)) and e
G(t0) is the
Euclidean metric, for the orthonormal frame {ei}m
i=1 with respect to η(t), Fi=
dF (ei) so that
Fi•Fi=e
G(t0) (dF (ei), dF (ej))
=e
G(t) (dF (ei), dF (ej)) + O(τ)
=η(ei, ej) + O(τ)
=δij +O(τ)
(2.20)
because dF is bounded uniformly with respect to e
Gand e
Gvaries smoothly in
t. Therefore,
m
2τ−ηij Fi•Fj
2τ≤C3.
(2.21)
On the other hand, if we denote ζ=F∗
tG(t0) and decompose F(as a vector
in RQ) using Euclidean metric e
G(t0):
(2.22) F=F⊥+aiFi
where ai=ζijF•Fj. Here F⊥is the normal part of Fon Ft(M) with
respect to the Euclidean metric. Since G(t)→G(t0) smoothly as t→t0,
ζij =δij +O(|t0−t|) and thus
|F|2=F•F= [F⊥+ (ζijF•Fi)·Fj]•[F⊥+ (ζklF•Fk)·Fl]
=|F⊥|2+
m
X
i=1
(Fi•F)2+O(τ)· |F|2.
(2.23)
We now simplify (∂t+ ∆η,euc)F•Ffurther by first noting that ∆η,eucF⊥
Ft(M) with respect to the Euclidean metric e
G(t0), and
∆η,eucFα= ∆η,
e
GFα+ηijΨα
βγ Fβ
iFγ
j
where Ψ = Γ
e
G(t)−Γeuc = Γ
e
G(t)−Γ
e
G(t0)=O(τ). Hence
∆η,eucF=e
H+E2
(2.24)
where E2=O(τ) because the energy density of Ftas a map from (M, η(t)) to
(X, G(t)) is mand G(t) is uniformly equivalent to G(t0) which is induced by
the Euclidean metric. Therefore using (2.16), we deduce
((∂t+ ∆η,euc)F)•F= (H+e
H+E2)•F
= (2 e
H+E1)•F⊥+O(τ)· |F|
(2.25)
where we have used ∆η,eucF⊥Ft(M), E2=O(τ), G(t)→G(t0) as t→t0
smoothly and E1∈N(X) in Uwith respect to G(t).
Rigidity of area non-increasing maps 11
Hence, we have
∂
∂t + ∆ηρ≤ρ C4+C5|F|2
τ−|F⊥|2
4τ2−(e
H+1
2E1)•F⊥
τ!.(2.26)
We are now in position to apply Wang’s argument [27]. We start by observ-
ing the Gaussian density is bounded from the C1bound of ft.
Claim 2.1.For any α > 0, there exists Cα>0 such that for all t→t0,
(2.27) ˆM
(t0−t)−m/2exp −|F(x, t)|2
α(t0−t)dvolη(t)(x)≤Cα.
Suppose the claim is true, first observe that
1
2ηij∂tηij =−|H|G2(t)−ηijJ(dF (∂i), dF (∂j))
≤ − e
G(t)(H, e
H) + C6
=−e
G(t)( e
H+E1,e
H) + C6
since J=∂tGis bounded and E1⊥Hwith respect to e
G(t). Combining with
(2.26), (2.15) and the fact that φ≥1 is bounded, we have
d
dt ˆM
φρ dvolη(t)
=ˆM∂tφ·ρ+φ·∂tρ+1
2φρ ·trη(∂tη)dvolη
=ˆM∂
∂t −∆ηφ·ρ+φ·∂
∂t + ∆ηρ dvolη
+1
2ˆM
φρ ·trη∂tη dvolη
≤ −aˆM
ρ|A|2ρ dvolη+C7ˆM
ρ(1 + |F|2
τ)dvolη
+ˆM
φρ−|F⊥|2
4τ2−(e
H+1
2E1)•F⊥
τ−e
G(t)( e
H+E1,e
H)dvolη.
(2.28)
On the other hand,
e
G(t)( e
H+E1,e
H)≥e
G(t0)( e
H+E1,e
H)−O(τ)|e
H+E1|euc|e
H|euc
≥(e
H+E1)•e
H−O(τ)(|e
H+1
2E1|2
euc +|E1|2
euc)
≥(1 −C9τ)|e
H+1
2E1|2
euc −C8
12 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
Hence
−|F⊥|2
4τ2−(e
H+1
2E1)•F⊥
τ−e
G(t)( e
H+E1,e
H)
≤ − |F⊥|2
4τ2−(e
H+1
2E1)•F⊥
τ−(1 −C9τ)|e
H+1
2E1|2
euc +C8
≤C10
|F|2
τ.
Therefore,
d
dt ˆM
φρ dvolη(t)≤C12 ˆM
ρ(1 + |F|2
τ)dvolη(t)≤C13
by the Claim 2.1. This shows that limt→t0´Mφρ dvolηexists by monotone
convergence Theorem. In the above argument, if we replace φby 1, we also
have
(2.29) d
dt ˆM
ρ dvolη≤C14.
So limt→t0´Mρ dvolηexists. Now we can follow the argument in [27] to deduce
(2.30) lim
t→t0ˆFt(M)
ρy0,t0dvol e
G(t)= 1.
It now follows from the proof of White’s regularity Theorem [28, Theorem 3.5]
as in [27] using (2.29) that (y0, t0) is a regular point.
It remains to prove Claim 2.1. Since Ftis given by graph of ft:M→N,
we might assume F(x, t) = (x, ft(x)) ∈M×Nembedded in RQ. Since Xis
isometrically embedded into RQand Xis compact, there exists C4>0 such
that
C−1
4deuc(p, q)≤dG(t0)(p, q)≤C4deuc(p, q)
for all p, q ∈X⊂RQ. Using also the fact that G(t) are uniformly equivalent
to G(t0) on X, we see that for all p, q ∈X⊂RQ,
C−1
4deuc(p, q)≤dG(t)(p, q)≤C4deuc(p, q).
Since G(t) = π∗
Mg(t)⊕π∗
Nh(t), we have
(2.31) dπ∗
Mg(t)(p, q)2≤dG(t)(p, q)2≤C4(deuc (p, q))2.
Furthermore on M, since Ft= Id ×ftand λi≤C1,
(2.32) gij ≤ηij =gij +fα
ifβ
jhαβ ≤(1 + C2
1)gij.
Rigidity of area non-increasing maps 13
Hence η(t) is uniformly equivalent to g(t) on M×[0, t0). Therefore,
ˆM
(t0−t)−m/2exp −d2
euc(F(x, t), y0)
α(t0−t)dvolη(t)(x)
≤C5ˆMt
(t0−t)−m/2exp −d2
G(t)(y, y0)
α(t0−t)!dvolπ∗
Mg(t)(y)
≤C5ˆM
(t0−t)−m/2exp −d2
g(t)(x, x0)
α(t0−t)!dvolg(t)(x).
(2.33)
Since g(t) has bounded geometry of infinity order, the integral is bounded
by constant depending a constant independent of t. This proves Claim 2.1.
3. Proof of the Main Theorems
In this section, will prove our main results. We will continue to work under
the setting in Section 2, and show that under certain assumption on G, both
the area non-increasing and strictly area decreasing will be preserved under
the mean curvature flow coupled with the evolving metric G(t).
3.1. Evolution equations under Uhlenbeck trick. We want to study the
evolution of Sand Θ. To simplify computation, we use the abstract vector
bundle method as in [26] which we refer it as Uhlenbeck trick. Let x0∈Mand
t0∈(0, T ] be fixed. Let {EA}m
A=1 be an orthonormal frame with respect to
η(t0) at x0∈Mso that dFt0(Ei) = eigiven by (2.12) at (x1, ft0(x1)) ∈M×N
where x1=φt0(x0). We extend EAaround x0∈Mby parallel transport with
respect to η(t0), so that ∇EA= ∆EA= 0 at (x0, t0) and {dFt(EA)}m
A=1 is an
orthonormal basis for t=t0. Consider the O.D.E:
(3.1) ∂tEk
A=−1
2ηjk Fα
iFβ
jJαβEi
A−ηjk HαAβ
ij GαβEi
A;
Ek
A(x, t0) = Ek
A(x)
where J=∂tG. Here we use the notation H=Hα∂αand Aα
ij =Fα
|ij to
denote the mean curvature vector and the second fundamental form of Ft(M)
in (M×N, G(t)). By our choice of endomorphism, direct computation shows
that for all t∈(0, T ],
(3.2) η(EA, EB) = δAB .
Now we are ready to derive evolution equations for Sand Θ using the
graphical frame (2.12) and the Uhlenbeck trick. More precisely, at each
(x0, t0)∈M×(0, T ], we choose the graphical frame so that (2.13) holds at
(x0, t0) and then we extend it locally around (x0, t0) in space-time using the
discussion above to obtain {Ei(x, t)}m
i=1 nearby (x0, t0)∈M×(0, T ].
In the following, we will write Sii =S(Ei, Ei) = 1−λ2
i
1+λ2
iand its conjugate
Cii =2λi
1+λ2
i. We also denote Kg
ip to be the sectional curvature of the two plane
spanned by ui, up, etc. Kg
pp = 0 by convention. In the setting of graph frame
14 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
(2.12) at a point, we always assume that λ1≥λ2≥ · · · ≥ λm≥0. Hence we
have
(3.3)
S11 ≤S22 ≤ · · · ≤ Smm ;
C11 ≥C22 ≥ · · · ≥ Cmm ;
Θ1221 ≤Θijji for all i6=j.
For any t≥0, we also define
(3.4) m(t) = inf
x∈M{smallest eigenvalue of Θ(x, t)}.
With the frame {Ei(x, t)}m
i=1 around (x0, t0), we might treat S(Ei, Ej) as a
locally defined function. We have the following evolution equation of S.
Lemma 3.1. For any (x0, t0)∈M×(0, T ], under the graphical frame {Ei}m
i=1
we have ∂
∂t −∆ηS(Ei, Ei)(x0,t0)=I+II +III(3.5)
where
(3.6)
I=
n
X
a=1
m
X
l=1
2(Sii +Saa)|Aa+m
il |2;
II =C2
ii ·
m
X
k=1
Kg
ik −λ2
kKh
ik
1 + λ2
k
;
III =1
2C2
ii ·(∂tgii −∂thii).
Here λ2
kKh
ik is understood to be zero if k > n.
Proof. It is a slight modification of [25, (3.7)]. Since we are working on an
evolving background, we include the proof for readers’ convenience. We start
with the evolution equation of the tensor S. Firstly using ∂tF=H=τ(F)
and sαβ =gαβ −hαβ , we have
∂tSij =Dt(Fα
iFβ
jsαβ )
=∇i∆ηFα·Fβ
j˜
Sαβ +∇j∆ηFβ·Fα
i˜
Sαβ +Fα
iFβ
j(∂tgαβ −∂thαβ )
(3.7)
while
∆ηSij = ∆ηFα
i·Fβ
j˜
Sαβ + ∆ηFβ
j·Fα
i˜
Sαβ + 2Fα
|ikFβ
|jl ηkl ˜
Sαβ.
(3.8)
Now we apply the Ricci identity of (M, η) so that
∆ηFα
i=∇i∆ηFα+Rp
iFα
p−ηkl ˜
Rδγε αFγ
kFδ
iFε
l
(3.9)
where Rand ˜
Rdenote the curvature of ηand Grespectively. On the other
hand, Gauss equation infers that
Riq =ηklRiklq =ηkl ˜
Rδγεσ Fδ
iFγ
kFε
lFσ
q−Aδ
ilAγ
kqGδγ +Aδ
iqAγ
klGδ γ
(3.10)
Rigidity of area non-increasing maps 15
Combining all, we arrive at
∂
∂t −∆ηSij =ηklSq
jAδ
ilAγ
kqGδγ +ηkl Sq
iAδ
jl Aγ
kq Gδγ
−Sq
jAδ
iqHγGδγ −Sq
iAδ
jq HγGδγ
+ (Gασ −ηpqFα
pFσ
q)ηklFβ
jFδ
iFγ
kFε
lsαβ ˜
Rδγεσ
+ (Gβσ −ηpq Fβ
pFσ
q)ηklFα
iFδ
jFγ
kFε
lsαβ ˜
Rδγεσ
−2Aα
ikAβ
jl ηkl sαβ +Fα
iFβ
j(∂tgαβ −∂thαβ).
(3.11)
Now we employ the bundle trick by considering S(Ei, Ei) for 1 ≤i≤m.
Using (2.13), (2.14), (3.1) and (3.11), at (x0, t0) it satisfies
∂
∂t −∆ηS(Ei, Ei)
=∂
∂t −∆ηS(Ei, Ei) + 2S(∂tEi, Ei)
=2(1 −λ2
i)
1 + λ2
i
|Aa+m
il |2+2(1 −λ2
a)
1 + λ2
a
|Aa+m
il |2
−2λi
1 + λ2
i
˜
R(˜
Ei,˜
Ek,˜
Ek,˜
Em+i) + 2λi
1 + λ2
i
˜
R(˜
Ei,˜
Ek,˜
Ek,˜
Em+i)
+h(∂tg−∂th)( ˜
Ei,˜
Ei)−Sii ·(∂tg+∂th) ( ˜
Ei,˜
Ei)i
=I+II +III.
(3.12)
For II, using the product structure of G(t), we have
˜
R(˜
Ei,˜
Ek,˜
Ek,˜
Em+i) = ˜
R(ui+λivi, uk+λkvk, uk+λkvk,−λiui+vi)
(1 + λ2
k)(1 + λ2
i)
=−λiRg(ui, uk, uk, ui) + λiλ2
kRh(vi, vk, vk, vi)
(1 + λ2
k)(1 + λ2
i)
=−λi
1 + λ2
i
·Kg
ik −λ2
kKh
ik
1 + λ2
k
.
(3.13)
The assertion on II follows since Cii =2λi
1+λ2
i.III is similar.
Similarly, Θ(Ei, Ej, Ek, El) is a locally defined function around (x0, t0). For
notation convenience, we just write
Θijkl = Θ(Ei, Ej, Ek, El)
defined locally around (x0, t0). The following is a slight modification of [26,
Lemma 3.1].
16 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
Lemma 3.2. For any (x0, t0)∈M×(0, T ], under the graphical frame {Ei}m
i=1,
if Θ1221 +α > 0at (x0, t0)for some α∈R, then
(Θ1221 +α)∂
∂t −∆ηΘ1221 +1
2|∇Θ1221|2
≥ − 2α(Θ1221 +α)|A|2+ 4α S11
m
X
k=1
|A1+m
1k|2+S22
m
X
k=1
|A2+m
2k|2!
+ (Θ1221 +α) [(2) + (3)]
where |A|is the norm of the second fundamental form and
(2) = C2
11
m
X
k=1
Kg
1k−λ2
kKh
1k
1 + λ2
k
+C2
22
m
X
k=1
Kg
2k−λ2
kKh
2k
1 + λ2
k
(3) = 1
2C2
11 (∂tg11 −∂th11) + 1
2C2
22 (∂tg22 −∂th22).
Here λ2
kKh
ik is understood to be zero if k > n.
Proof. By In the following, let Eibe the graph frame at a point, which has
been extended to a frame using Uhlenbeck’s trick, we write Sii =S(Ei, Ei)
and Θijj i = Θ(Ei, Ej, Ej, Ei) etc. By Lemma 3.1, at (x0, t0we have
∂
∂t −∆ηΘ1221 =∂
∂t −∆η(S11 +S22)
=(1) + (2) + (3)
(3.14)
where
(1) =
n
X
a=1
m
X
l=1
2(S11 +Saa)|Aa+m
1l|2+
n
X
a=1
m
X
l=1
2(S22 +Saa)|Aa+m
2l|2
(2) = C2
11
m
X
k=1
Kg
1k−λ2
kKh
1k
1 + λ2
k
+C2
22
m
X
k=1
Kg
2k−λ2
kKh
2k
1 + λ2
k
(3) = 1
2C2
11 (∂tg11 −∂th11) + 1
2C2
22 (∂tg22 −∂th22).
Rigidity of area non-increasing maps 17
Since Sii +Sjj ≥Θ1221 for all 1 ≤i6=j≤m, (1) can be estimated as:
(1) = X
1≤k≤m;a6=1,2
2(S11 +Saa)|Aa+m
1k|2+
m
X
k=1
2(S11 +S22)|A2+m
1k|2
+X
1≤k≤m;a6=1,2
2(S22 +Saa)|Aa+m
2k|2+
m
X
k=1
2(S11 +S22)|A1+m
2k|2
+ 4S11
m
X
k=1
|A1+m
1k|2+ 4S22
m
X
k=1
|A2+m
2k|2
≥2Θ1221 X
1≤k≤m;a6=1,2
(|Aa+m
1k|2+|Aa+m
2k|2) +
m
X
k=1
(|A2+m
1k|2+|A1+m
2k|2)!
+ 4S11
m
X
k=1
|A1+m
1k|2+ 4S22
m
X
k=1
|A2+m
2k|2.
On the other hand, for each 1 ≤k≤m
∇EkΘ(E1, E2, E2, E1) = −4A1+m
1kλ1
1 + λ2
1
+A2+m
2kλ2
1 + λ2
2
=−2(C11A1+m
1k+C22A2+m
2k)
so that
|∇Θ1221|2= 4
m
X
k=1
(C11A1
1k+C22A2
2k)2.
On the other hand, by the proof of [26, Lemma 3.1],
2Θijj iSii = Θ2
ijj i +C2
jj −C2
ii
for all i6=j. In fact,
Θ2
ijj i +C2
jj −C2
ii =Θ2
ijj i +2(λ2
j−λ2
i)Θijj i
(1 + λ2
i)(1 + λ2
j)
=Θijj i ·2(1 −λ2
iλ2
j+λ2
j−λ2
i)
(1 + λ2
i)(1 + λ2
j)
=2Θijj iSii .
18 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
Hence we have
m
X
k=1
Θ1221 4S11(A1+m
1k)2+ 4S22(A2+m
2k)2+1
2|∇Θ1221|2
=2
m
X
k=1
|A1+m
1k|2Θ2
1221 +C2
22 −C2
11
+ 2
m
X
k=1
|A2+m
2k|2Θ2
1221 +C2
11 −C2
22+ 2
m
X
k=1
(C11A1+m
1k+C22A2+m
2k)2
=2Θ2
1221
m
X
k=1 |A1+m
1k|2+|A2+m
2k|2+ 2
m
X
k=1
(C22A1+m
1k+C11A2+m
2k)2
≥2Θ2
1221
m
X
k=1 |A1+m
1k|2+|A2+m
2k|2.
Therefore,
(Θ1221 +α)(1) + 1
2|∇Θ1221|2
≥2Θ1221(Θ1221 +α) X
1≤k≤m;a6=1,2
(|Aa+m
1k|2+|Aa+m
2k|2) +
m
X
k=1
(|A2+m
1k|2+|A1+m
2k|2)!
+ (Θ1221 +α)"4S11
m
X
k=1
(A1+m
1k)2+ 4S22
m
X
k=1
(A2+m
2k)2#+1
2|∇Θ1221|2
≥2Θ1221(Θ1221 +α) X
1≤k≤m;a6=1,2
(|Aa+m
1k|2+|Aa+m
2k|2) +
m
X
k=1
(|A2+m
1k|2+|A1+m
2k|2)!
+ 2Θ2
1221
m
X
k=1 |A1+m
1k|2+|A2+m
2k|2+ 4α"S11
m
X
k=1
(A1+m
1k)2+S22
m
X
k=1
(A2+m
2k)2#
≥ −2α(Θ1221 +α)
m
X
k=1
n
X
a=1
(|Aa+m
1k|2+|Aa+m
2k|2)
+ 4α"S11
m
X
k=1
(A1+m
1k)2+S22
m
X
k=1
(A2+m
2k)2#.
because (Θ1221 +α)Θ1221 ≥ −α(Θ1221 +α). Putting this back to (3.14) yields
the result.
In case Θ >0, we have the following which is obtained in [26] and has been
used in the proof of long time existence in §2.3.
Corollary 3.1. As in Lemma 3.2, if Θ>0, then
Rigidity of area non-increasing maps 19
∂
∂t −∆ηlog det Θ
det(1
2η?η)≥a|A|2−C
for some a > 0and some constant Cdepending only on the bounds of the
curvatures of g(t), h(t), the bounds of ∂tg, ∂th,m, n and the positive lower
bound of Θ.
Proof. Since Θ >0, we can take α= 0 in the proof of Lemma 3.2 before
applying (Θ1221 +α)Θ1221 ≥ −α(Θ1221 +α). The result follows.
3.2. Monotonicity in static background. In this subsection, we consider
the case when G(t) is static in tand will prove the preservation of area non-
increasing under various curvature conditions. Let (Mm, g),(Nn, h) be smooth
compact manifolds. Let ℓ= min{m, n} ≥ 2.
(A): g(t) = gand h(t) = hare time-independent satisfying
(3.15) (Ricg
min −Rich
max + (m−ℓ)·κM+ (n−ℓ)·κN≥0,and
κM+κN≥0
(B): g(t) = gand h(t) = hare time-independent satisfying
(3.16) κM≥0 and τN≤2(m−ℓ) + ℓ−1
ℓ−1κM
Here κMis the lower bounds of the sectional curvature of gand τNis the upper
bound of the sectional curvature of h.
We want to estimate (2) + (3) in Lemma 3.2 under the above conditions.
Note that (3) is always zero in static case.
Lemma 3.3. With the same assumptions and notations as in Lemma 3.2, we
have the following:
(i) Under the condition (A), we have
(2) + (3)≥1
2
ℓ
X
p=3 C2
11(Kg
1p+Kh
1p) + C2
22(Kg
1p+Kh
1p)Spp
+ (Kg
12 +Kh
12)(λ2
1+λ2
2)
(1 + λ2
1)(1 + λ2
2)Θ1221.
(ii) Under condition (B), we have
(2) + (3)≥(κM+τN)"1
2C2
11 +C2
22ℓ
X
p=3
Spp +λ2
1+λ2
2
(1 + λ2
1)(1 + λ2
2)Θ1221#.
Proof. (i) Assume (A). Then (3) = 0 and
(2) =C2
11
m
X
p=1
Kg
1p−λ2
pKh
1p
1 + λ2
p
+C2
22
m
X
p=1
Kg
2p−λ2
pKh
2p
1 + λ2
p
20 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
where
2
m
X
p=1
Kg
1p−λ2
pKh
1p
1 + λ2
p
=
m
X
p=1
Kg
1p(1 + Spp)−
ℓ
X
p=1
Kh
1p(1 −Spp)
≥Ricg
11 −Rich
11 +
ℓ
X
p=2
(Kg
1p+Kh
1p)Spp + (m−ℓ)κM+ (n−ℓ)κN
≥
ℓ
X
p=3
(Kg
1p+Kh
1p)Spp + (Kg
12 +Kh
12)S22.
Similarly,
2
m
X
p=1
Kg
2p−λ2
pKh
2p
1 + λ2
p
≥
ℓ
X
p=3
(Kg
2p+Kh
2p)Spp + (Kg
12 +Kh
12)S11 .
On the other hand, since
C2
11S22 +C2
22S11 =2(λ2
1+λ2
2)
(1 + λ2
1)(1 + λ2
2)Θ1221,
we conclude that (i) is true.
(ii) Assume (B). Observe that τNcan be assumed to be nonnegative. In
this case,
2
m
X
p=1
Kg
1p−λ2
pKh
1p
1 + λ2
p
≥
m
X
p=2
2κM
1 + λ2
p
−τN
n
X
p=2
2λ2
p
1 + λ2
p
=
m
X
p=2
(1 + Spp)κM−τN
ℓ
X
p=2
(1 −Spp)
=
ℓ
X
p=2
(κM+τN)Spp + (ℓ−1)(κM−τN) + 2(m−ℓ)κM
≥
ℓ
X
p=3
(κM+τN)Spp + (κM+τN)S22
Similarly,
2
m
X
p=1
Kg
2p−λ2
pKh
2p
1 + λ2
p
≥
ℓ
X
p=3
(κM+τN)Spp + (κM+τN)S11
Rigidity of area non-increasing maps 21
so that
(2)≥1
2C2
11 +C2
22ℓ
X
p=3
(κM+τN)Spp
+1
2(κM+τN)C2
11S22 +C2
22S11
=(κM+τN)"1
2C2
11 +C2
22ℓ
X
p=3
Spp +λ2
1+λ2
2
(1 + λ2
1)(1 + λ2
2)Θ1221#.
As before, one can conclude that (ii) is true.
Recall that for t≥0,
m(t) = inf
x∈M{smallest eigenvalue of Θ(x, t)}.
Now we are ready to prove the Case (A) and (B) in our main Theorem.
Theorem 3.1. Let (Mm, g),(Nn, h)be two compact manifolds. Suppose f0
is a smooth map from Mto N. Let F:M→(M×N, g ⊕h)be a smooth
mean curvature flow defined on M×[0, T )with 0< T ≤+∞and initial map
F0= Id ×f0. Moreover, assume Fis a graph given by a map ft:M→Nfor
all t∈[0, T ). Assume that f0:M→Nis area non-increasing and one of the
condition (A) or (B) holds, then
(i) ftis area non-increasing for t∈[0, T );
(ii) If f0is strictly area decreasing at a point, then m(t)>0for t > 0and
is nondecreasing in twhere m(t)is defined in (3.4). In particular, F
has long time solution. Moreover, if κM+κN>0in condition (A) or
κM>0in condition (B), then f0is homotopically trivial.
(iii) If m, n ≥3, and κM+κN>0in condition (A) or κM>0in condition
(B), then either f0homotopically trivial or f0is a Riemanian submer-
sion (if m > n), local isometry (if m=n), isometric immersion (if
m < n).
Before we prove the theorem, we want to point out that the results on
f0being homotopically trivial under the assumption that f0is strictly area
decreasing have been obtained by Lee-Lee [16] and Tsai-Tsui-Wang [26]. We
just slightly generalize their results to assume that f0is strictly area decreasing
at a point.
Proof of Theorem 3.1. We only prove the case when (M, g),(N, h) satisfy con-
dition (A). The proof is similar if they satisfy condition (B). We may assume
that Fis smooth on M×[0, T ]. To prove (i) and (ii), let 0 ≤ϕ0≤1 be a
22 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
smooth function on Mand let ϕbe the solution of the heat equation:
(3.17)
∂
∂t −∆ηϕ= 0,on M×[0, T ];
ϕ=ϕ0at t= 0.
By the maximum principle and the strong maximum principle, we have 1 ≥
ϕ≥0 and ϕ > 0 for t > 0 if ϕ0is positive somewhere. Consider the following
perturbation of Θ:
(3.18) ˜
Θ = Θ −1
2eϕ·η?η
where ε, L, δ > 0, with 0 < ε < 1,0< δ < 1 and
(3.19) eϕ=δe−Lt ϕ2−εeL(t−T).
where L > 0 is to be chosen. Define
(3.20) e
m(t) = inf
x∈M{smallest eigenvalue of e
Θ(x, t)}.
Suppose e
m(0) >0 and suppose e
m(t)<0 for some t > 0. Then there is 0 <
t0≤Tsuch that e
m(t0) = 0 and e
m(t)>0 for 0 ≤t < t0. Hence by (3.3), there
is x0∈Mand in the graph frame at (x0, t0), e
Θ(E1, E2, E2, E1) = 0 = e
m(t0).
Extend Eiusing Uhlenbeck’s trick to an open set in spacetime near (x0, t0)
following the discussion in subsection 3.1. Then we have
(3.21) ∂
∂t −∆η(e
Θ(E1, E2, E2, E1)) ≤0;
∇Θ1221 −2δe−Lt0ϕ∇ϕ=∇e
Θ(E1, E2, E2, E1) = 0.
Here and below, we write e
Θ(E1, E2, E2, E1) as e
Θ1221 etc. Since e
Θippi ≥e
Θ1221
for i= 1,2 and p≥3. Hence if p≥3, then Spp ≥1
2Θ1221 =1
2(e
Θ1221 +eϕ).
Moreover, Θ1221 =eϕat (x0, t0).
Rigidity of area non-increasing maps 23
Let α > 0 to be determined later such that Θ1221 +α > 0. by Lemma 3.2,
Lemma 3.3, (3.21) implies:
1
2|∇Θ1221|2
≥(Θ1221 +α)∂
∂t −∆ηe
Θ1221 +1
2|∇Θ1221|2
=(Θ1221 +α)∂
∂t −∆ηΘ1221 +L(eϕ+ 2εeL(t0−T)) + 2δe−Lt0|∇ϕ|2+1
2|∇Θ1221|2
≥ − 2α(Θ1221 +α)|A|2+ 4α S11
m
X
p=1
(A1+m
1p)2+S22
m
X
p=1
(A2+m
2p)2!
+ (Θ1221 +α)ℓ
X
p=3 2λ2
1
(1 + λ2
1)2(Kg
1p+Kh
1p) + 2λ2
2
(1 + λ2
2)2(Kg
2p+Kh
2p)Spp
+ (Kg
12 +Kh
12)(λ2
1+λ2
2)
(1 + λ2
1)(1 + λ2
2)Θ1221
+ (Θ1221 +α)L(eϕ+ 2εeL(t0−T)) + 2δe−Lt0|∇ϕ|2.
(3.22)
Hence if we let β=δe−Lt0ϕ2+εeL(t0−T)and α= 4β, then Θ1221 =δe−Lt0ϕ2−
εeL(t0−T)and so Θ1221 +α≥3β > 0 at (x0, t0). And ∇Θ1221 = 2δe−Ltϕ∇ϕat
(x0, t0). Hence,
0≥ − C1β2+ 16β S11
m
X
p=1
(A1+m
1p)2+S22
m
X
p=1
(A2+m
2p)2!
+ (Θ1221 +α)ℓ
X
p=3 2λ2
1
(1 + λ2
1)2(Kg
1p+Kh
1p) + 2λ2
2
(1 + λ2
2)2(Kg
2p+Kh
2p)Spp
+ (Kg
12 +Kh
12)(λ2
1+λ2
2)
(1 + λ2
1)(1 + λ2
2)Θ1221
+ 3βLβ + 2δe−Lt0|∇ϕ|2−2δ2e−2Lt0ϕ2|∇ϕ|2
(3.23)
where |A|is the norm of the second fundamental form and C1is a constant
depending only on the upper bounds of |A|, λi, m, n which is independent of
L, δ, ε. By (3.21), we have for 1 ≤p≤ℓ,
0 =∇Epe
Θ1221
=∇EpΘ1221 −2δe−Lt0ϕϕp
=−4 A1+m
1pλ1
1 + λ2
1
+A2+m
2pλ2
1 + λ2
2!−2δϕe−Lt0ϕp.
24 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
Claim:λ2
1≥1
3. In fact, since λ1≥λ2, we have
2S11 ≤S11 +S22 = Θ1221 =eϕ≤δ≤1.
From this one can see that λ2
1≥1
3and hence
|(A1+m
1p)2−(A2+m
2p)2| ≤2|A||A1+m
1p−A2+m
2p|
≤C2δϕe−Lt0|∇ϕ|+1−λ2(1 + λ2
1)
λ1(1 + λ2
2)
≤C2δϕe−Lt0|∇ϕ|+β.
for some constant C2>0 depending only on the upper bounds of |A|, λi. Here
we have used the Claim that λ1≥1
3and the fact that
1−λ2(1 + λ2
1)
λ1(1 + λ2
2)=
(1 + λ2
2)(λ1−λ2)(1 −λ1λ2)
λ1(1 + λ2
2)≤C|Θ1221| ≤ Cβ
for some constant depending only on the upper bound of λi. Hence
S11(A1+m
1p)2+S22(A2+m
2p)2=S11 (A1+m
1p)2−((A2+m
2p)2+ Θ1221(A2+m
2p)2
≥ − C3δϕe−Lt0|∇ϕ|+β
for some constant C3depending only on the upper bounds of |A|, λi. Com-
bining this with (3.22), using the facts that Spp ≥1
2Θ1221 =−1
2eϕ≥ −1
2βfor
p≥3 and κM+κN>0, we have
0≥ − C4β2+βδe−Lt0ϕ|∇ϕ|+ 3βLβ + 2δe−Lt0|∇ϕ|2−2δ2e−2Lt0ϕ2|∇ϕ|2
for some C4>0 depending only on the upper bounds of |A|, λi, m, n and the
curvatures of g, h. Now
6βδe−Lt0|∇ϕ|2≥6δ2e−2Lt0ϕ2|∇ϕ|2
and
C4βδe−Lt0ϕ|∇ϕ| ≤ δ2e−2Lt0ϕ2|∇ϕ|2+C2
4
4β2.
This implies that
0≥ −C5β2+ 3Lβ2.
for some C5>0 depending only on the upper bounds of |A|, λi, m, n and the
curvatures of g, h. This is a contradiction if we choose L=C5. Namely for
this choice of L,e
Θ>0 for t > 0 provided e
Θ>0 at t= 0.
Suppose Θ ≥0 initially, we let ϕ0= 0. Then the above result implies that
Θ + εeL(t−T)≥0 for all t > 0. Let ε→0, we conclude that (i) is true.
To prove (ii), suppose f0is strictly increasing at some point x. Then we can
find a smooth function 1 ≥ϕ0≥0 so that ϕ0>0 at xand Θ −1
2ϕ0η?η≥0
initially. Let ϕbe the solution to (3.17) and let L=C5be as above and
1> δ, ε > 0. Let eϕbe as in (3.19) Then we can conclude that Θ−1
2eϕη ?η≥0
for all t > 0. Let ε→0, since ϕ > 0 at t > 0 we conclude that Θ >0 for
t > 0. This proves the first part of (ii).
Rigidity of area non-increasing maps 25
In order to prove m(t) is non-decreasing, it is sufficient to prove that it is
non-decreasing on [t, T ] for all t > 0. Since Θ >0 for t > 0, without loss of
generality, we may assume Θ ≥ρ0>0 at t= 0 and to prove that Θ ≥ρ0for
t > 0. For any ε > 0 consider Θ = Θ + 1
2εtη ?η. We claim that the infimum
of the eigenvalues of Θ is attained at t= 0. Otherwise, there is t0>0 and
x0∈Msuch that Θ1221 attains the infimum in the graph frame. Using the
extension with Uhlenbeck’s trick, we have
∂
∂t −∆ηΘ1221 ≤0; ∇Θ1221 = 0
at (x0, t0). Since Θ >0 everywhere, one obtain from as in (3.22) with α= 0,
0≥∂
∂t −∆ηΘ1221
≥ℓ
X
p=3 2λ2
1
(1 + λ2
1)2(Kg
1p+Kh
1p) + 2λ2
2
(1 + λ2
2)2(Kg
2p+Kh
2p)Spp
+ (Kg
12 +Kh
12)(λ2
1+λ2
2)
(1 + λ2
1)(1 + λ2
2)Θ1221+ε
≥ε
(3.24)
because Spp ≥1
2Θ1221 >0 for p≥3 and κM+κN≥0. This is impossible. Let
ε→0, we conclude that m(t)≥ρ0for t≥0. From this we also conclude that
Fhas long time solution which is a graph for all time by Theorem 2.1.
Suppose κM+κN>0. Let b
Θ = e−atΘ where a > 0 to be determined.
Suppose the infimum of the eigenvalues of b
Θ in M×[0, T ] is attained at some
spacetime point (x0, t0) with t0>0. Then as in (3.24), at this point,
0≥∂
∂t −∆ηb
Θ1221
≥(κM+κN)(λ2
1+λ2
2)
(1 + λ2
1)(1 + λ2
2)b
Θ1221 −ab
Θ1221.
(3.25)
Since Θ ≥ρ0for all t, we conclude that λi≤(2
ρ0−1) in spacetime. Since λ2
iare
uniformly bounded in space and time, there is a constant C6>0 depending
only on ρ0,
(λ2
1+λ2
2)
(1 + λ2
1)(1 + λ2
2)≥C62λ2
1
1 + λ2
1
+2λ2
1
1 + λ2
1
=C6(2 −Θ1221)
≥C6(2 −m∞).
26 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
where m∞= limt→∞ m(t) which exists and finite because m(t) is nondecreasing
and is bounded above by 2. If m∞<2, then (3.25) implies:
0≥C7Θ1221 −aΘ1221
for some constant C7>0 depending only on C6, κM+κN>0 and m∞<2.
Choose a=1
2C7, we have a contradiction because Θ1221 >0. Hence for this
choice of a, by letting T→+∞we have e−atm(t)≥m(0) ≥ρ0>0 for all t.
This is impossible. Hence m∞= 2 and f0is homotopically trivial.
To prove (iii), we claim the following: for t > 0, suppose at a point Θ1221 = 0,
then λi= 0 for all 1 ≤i≤ℓ. This is true, for t > 0 either there is a point with
Θ>0 which implies f0is homotopically trivial, or λi= 1 for 1 ≤i≤ℓ. Hence
either f0is homotopically trivial, if we let t→0, we conclude that λi= 1 for
1≤i≤ℓat t= 0. This implies that f0is as described in the theorem.
To prove the claim, suppose Θ1221 = 0 at (x0, t0) with t0>0. By (i), Θ ≥0
in spacetime, we have ∂
∂t −∆ηΘ1221 = 0,∇Θ1221 = 0. Then by (3.22) for
α > 0 with ϕ= 0, ε = 0, we have
0≥(Θ1221 +α)∂
∂t −∆ηΘ1221 +1
2|∇Θ1221|2
≥ − 2α2|A|2+ 4α S11
m
X
p=1
(A1+m
1p)2+S22
m
X
p=1
(A2+m
2p)2!
+α
ℓ
X
p=3 2λ2
1
(1 + λ2
1)2(Kg
1p+Kh
1p) + 2λ2
2
(1 + λ2
2)2(Kg
2p+Kh
2p)Spp
because Θ1221 = 0. Divide by αand then let α→0, we have
0≥4 S11
m
X
p=1
(A1+m
1p)2+S22
m
X
p=1
(A2+m
2p)2!
+ (κM+κN)
ℓ
X
p=3 2λ2
1
(1 + λ2
1)2+2λ2
2
(1 + λ2
2)2Spp
(3.26)
Since Θ1221 = 0 and λ1≥λ2, we have λ1≥1. Using the fact that ∇Θ1221 = 0,
as before,
|(A1+m
1p)2−(A2+m
2p)2| ≤2|A||A1+m
1p−A2+m
2p|
≤2|A||1−λ2(1 + λ2
1)
λ1(1 + λ2
2)|
=0.
Rigidity of area non-increasing maps 27
Hence S11
m
X
p=1
(A1+m
1p)2+S22
m
X
p=1
(A2+m
2p)2!=S11 (A1+m
1p)2−(A2+m
2p)2
=0.
(3.26) implies:
0≥(κM+κN)
ℓ
X
p=3 2λ2
1
(1 + λ2
1)2+2λ2
2
(1 + λ2
2)2Spp.
Since Spp ≥1
2(S11 +S22) = 0, λ1≥1 and κM+κN>0, we have λp= 1 for
p≥3. Hence λ1≥λ2≥λp≥1 and 1 −λ2
1λ2
2= 0, we have λp= 1 for all
1≤p≤ℓ. This completes the proof of the theorem under condition (A). The
case (B) is similar.
Here are some applications.
Corollary 3.2. Suppose Nis either Sn, n ≥3,CPn/2, n ≥4or HPn/4, n ≥4
with the standard metrics. If (Mn, g)is a compact manifold with nonnegative
sectional curvature such that RicM≥RicN. Then any area non-increasing
map from Mto Nwith nonzero degree must be an isometry.
Proof. Since f0has nonzero degree, it is not homotopically trivial. By the
theorem, we conclude that f0is a local isometry and hence is an isometry
because Nis simply connected.
3.3. Monotonicity in evolving background. In this subsection, we con-
sider the case when G(t) is evolving. This will eventually be applied to the
case of non-negative 1-isotropic curvature. We first introduce the following
definition.
Definition 3.1. For a Riemannian manifold, at a point, Ric3(π)is the Ricci
curvature tensor restricted on a three dimensional subspace πof the tangent
space. We say that Ric3≥aat a point if Ric3(π)≥afor all π. We say that
Ric3≥aon Mif it is true at all points. In this case, ais called a lower bound
of Ric3.
We consider the following two situations:
(C): ∂tg=−Ricg, ∂th=−Richon [0, T ] where
(3.27) χg(t) + χh(t)≥0; and (m−ℓ)·χg(t) + (n−ℓ)·χh(t)≥0
for all t∈[0, T ].
(D): ∂tg=−Ricg, ∂th= 0 on [0, T ] where
(3.28) χg(t)≥0; and τN≤0
for all t∈[0, T ]. Here χg(t), χh(t) denote the infimum of Ric3(g(t)),Ric3(h(t))
and τNis the upper bound of the sectional curvature of h.
28 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
Lemma 3.4. With the same assumptions and notations as in Lemma 3.2, we
have the following:
(i) Under condition (C),
(2) + (3)≥ −C|Θ1221|+2λ2
1
(1 + λ2
1)2
ℓ
X
p=3
(Kg
1p+Kg
2p+Kh
1p+Kh
2p)Spp;
(ii) Under condition (D),
(2) + (3)≥ −C|Θ1221|+2λ2
1
(1 + λ2
1)2 ℓ
X
p=3
(Kg
1p+Kg
2p)Spp −
ℓ
X
a=1
2τNλ2
a
1 + λ2
a!
for some constant C > 0depending only on the bounds of the curvatures of
g(t), h(t), m, n.
Proof. (i) Assume (C). Then
(2) + (3) = 2λ2
1
(1 + λ2
1)2 m
X
p=1
2(Kg
1p−λ2
pKh
1p)
1 + λ2
p
−Ricg
11 + Rich
11!
+2λ2
2
(1 + λ2)2 m
X
p=1
2(Kg
2p−λ2
pKh
2p)
1 + λ2
p
−Ricg
22 + Rich
22!
where
m
X
p=1
2(Kg
1p−λ2
pKh
1p)
1 + λ2
p
−Ricg
11 + Rich
11
=
m
X
p=1
Kg
1p(1 + Spp)−
ℓ
X
p=1
Kh
1p(1 −Spp)−
m
X
p=1
Kg
1p+
n
X
p=1
Kh
1p
=(Kg
12 +Kh
12)S22 +
ℓ
X
p=3
(Kg
1p+Kh
1p)Spp +
m
X
p=ℓ+1
Kg
1p+
n
X
p=ℓ+1
Kh
1p.
Similarly,
m
X
p=1
2(Kg
2p−λ2
pKh
2p)
1 + λ2
p
−Ricg
22 + Rich
22
=(Kg
12 +Kh
12)S11 +
ℓ
X
p=3
(Kg
2p+Kh
2p)Spp +
m
X
p=ℓ+1
Kg
2p+
n
X
p=ℓ+1
Kh
2p
On the other hand,
2λ2
1
(1 + λ2
1)2−2λ2
2
(1 + λ2
2)2=(λ2
1−λ2
2)Θ1221
(1 + λ2
1)(1 + λ2
2).
Rigidity of area non-increasing maps 29
Hence,
(2) + (3) = 2λ2
2
(1 + λ2
2)2−2λ2
1
(1 + λ2
1)2×
(Kg
12 +Kh
12)S11 +
ℓ
X
p=3
(Kg
2p+Kh
2p)Spp +
m
X
p=ℓ+1
Kg
1p+
n
X
p=ℓ+1
Kh
1p
+2λ2
1
(1 + λ2
1)2(Kg
12 +Kh
12)Θ1221 +
ℓ
X
p=3
(Kg
1p+Kg
2p+Kh
1p+Kh
2p)Spp
+
m
X
p=ℓ+1
(Kg
1p+Kg
2p) +
n
X
p=ℓ+1
(Kh
1p+Kh
2p)
≥ − C|Θ1221|+2λ2
1
(1 + λ2
1)2
ℓ
X
p=3
(Kg
1p+Kg
2p+Kh
1p+Kh
2p)Spp
for some positive constants Cdepending only on the bounds of the curvatures
of g(t), h(t), m, n. This completes the proof of (i).
(ii) Assume (D), then
(2) + (3) = 2λ2
1
(1 + λ2
1)2 m
X
p=1
2(Kg
1p−λ2
pKh
1p)
1 + λ2
p
−Ricg
11!
+2λ2
2
(1 + λ2)2 m
X
p=1
2(Kg
2p−λ2
pKh
2p)
1 + λ2
p
−Ricg
22!
Now
m
X
p=1
2(Kg
1p−λ2
pKh
1p)
1 + λ2
p
−Ricg
11
≥
m
X
p=1
Kg
1p(1 + Spp)−
ℓ
X
a=1
2τNλ2
a
1 + λ2
a
−Ricg
11
≥Kg
12S22 +
ℓ
X
p=3
Kg
1pSpp −
ℓ
X
a=1
2τNλ2
a
1 + λ2
a
Similarly,
m
X
p=1
2(Kg
2p−λ2
pKh
2p)
1 + λ2
p
−Ricg
22
≥Kg
12S11 +
ℓ
X
p=3
Kg
2pSpp −
ℓ
X
a=1
2τNλ2
a
1 + λ2
a
.
Hence as in the proof of (i), one can conclude that (ii) is true.
30 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
We first show that the area non-increasing is preserved as long as the ambient
space is evolving and the correspond Rigidity. More precisely, we have the
following.
Theorem 3.2. Let (Mm, g(t)),(Nn, h(t)) be two compact manifolds with ℓ=
min{m, n} ≥ 3,t∈[0, T ]. Suppose f0is a smooth map from Mto N. Let
F:M→(M×N, g ⊕h)be a smooth mean curvature flow defined on M×[0, T ]
with initial map F0= Id ×f0. Moreover, assume Fis a graph given by a map
ft:M→Nfor all t. Assume that f0is with area non-increasing from
(M, g(0)) to (N, h(0)). Suppose (C) or (D) is true, then the followings are
true.
(i) ftis area non-increasing for t∈[0, T ].
(ii) If f0is strictly area decreasing at a point, then ftis strictly area de-
creasing for t > 0. Moreover, there is a constant a > 0depending only
on the bounds of the curvatures of g(t), h(t), m, n so that eatm(t)is
non-decreasing. In particular, if g(t), h(t)are defined on M×[0, Tmax)
and N×[0, Tmax)respectively, then the mean curvature flow Fexists
and remains graphic on M×[0, Tmax).
(iii) If in addition, χg+χh>0in case of (C) and; χg>0in case of
(D), then either ftis strictly area decreasing for all t∈(0, T ]or f0
is a Riemanian submersion (if m > n), local isometry (if m=n),
isometric immersion (if m < n).
(iv) If in addition τN<0in case of (D), then ftis strictly area decreasing
for all t∈(0, T ].
Proof. We only prove the case (C) while the case (D) can be proved using
similar argument. Let φ, ϕ be as in the proof the Theorem 3.1 with 0 ≤δ, ε < 1
and L > 0 to be determined. The proof of (i) and the first statement of (ii)
are similar to the proof of Theorem 3.1.
We focus on the second assertion of (ii). Let a > 0 to be determined and let
Θ = eatΘ. We want to prove that eatm(t) is nondecreasing. Since Θ >0 for
t > 0, we may assume that m(0) >0. Suppose eatm(t)<m(0) for some t > 0.
Then there exists x0∈M, t0>0 such that eat0Θ1221 =eat0m(t0)≤eatm(t) for
0≤t≤t0. Then we by Lemmas 3.2 (with α= 0) and Lemma 3.4,
0≥∂
∂t −∆ηΘ1221 +1
2Θ−1
1221|∇Θ1221|2
≥ − C4Θ1221 +aΘ1221
for some constant C4depending only on the bounds of the curvatures of
g(t), h(t), m, n, at some point in space-time. This is impossible, if we take
a= 2C4because Θ >0. Hence eatm(t) is nondecreasing. By Theorem 2.1, the
last assertion of (ii) is true.
The proof of (iii) and (iv) are similar to the proof of Theorem 3.1(iii) using
the fact that if Θ1221 = 0 then λ1≥1.
Rigidity of area non-increasing maps 31
Remark 3.1.The conditions (C) and (D) are not necessarily preserved along
the Ricci flow. When n= 3, χg(t)≥0 is equivalent to Ric ≥0 which is
preserved along the Ricci flow by Hamilton [19]. When n≥4, χg(t)≥0 can
be ensured by χIC1≥0, which is preserved along the Ricci flow thanks to the
work of Brendle-Schoen [4] and Nguyen [18].
Now we are ready to prove the rigidity of maps under non-negative 1-
isotropic curvature condition. We need the following:
Lemma 3.5. Suppose (Mn, g0), n ≥3is compact, simply connected, non-
symmetric, irreducible compact manifold such that χI C 1(g0)≥0. Let g(t), t ∈
[0, Tmax)be the maximal Ricci flow solution starting from g0. Then Tmax <∞
and as t→Tmax, the curvature of g(t)will tend to infinity. In particular, if g0
is Einstein, then g0has positive sectional curvature. When n= 3,χI C 1≥0is
understood to be Ric ≥0.
Proof. If n= 3, then the result follows from the work of Hamilton [19, 20]
because (M, g0) is locally irreducible.
Suppose n≥4. χIC1(g0)≥0 implies that the scalar curvature R(g0)≥0.
If R(g0)≡0, then g0is flat. In fact, for an orthonormal frame ei, let Kij with
i6=jbe the sectional curvature of the two planes spanned by ei, ej. Then we
have Kik +Kil = 0 for all i < k < l. Hence Kij = 0 for all i < j. Hence we
must have R(g0)>0 somewhere. By the evolution equation of R(g(t)) and
the strong maximum principle, R(g(t)) >0 for t > 0. Hence Tmax <∞.
First, for n≥4, it follows from [4, 18] that χIC 1(g(t)) ≥0 for all t∈
[0, Tmax). Suppose χIC1(g0)>0 at some point, then it follows from [4, 18]
that χIC 1(g(t)) >0 for all t > 0 and the result follows from the work of
Brendle [5].
From now on, we assume this is not the case. We might assume g=g(t0) is
irreducible by choosing t0sufficiently small. We apply the Berger classification
Theorem to deduce that (Mn, g) is either quaternion-K¨ahler or has holonomy
group SO(n) or U(n
2) since the remaining are Ricci flat and hence flat by χIC 1≥
0. It also follows from [6] that quaternion-K¨ahler case is indeed symmetric and
so does g0. Hence, the quaternion-K¨ahler case is ruled out.
Suppose Hol(M, g) = SO(n), we claim that we must have χIC1(g)>0. This
was implicitly proved in [3], we include it for readers’ convenience. Suppose
there is x0∈M,λ∈[0,1] and orthonormal frame {ei}4
i=1 at (x0, t0) such that
(3.29) R1331 +λ2R1441 +R2332 +λ2R2442 −2λR1234 = 0.
By [3, Proposition 5], the equality is invariant under parallel transport. Since
Hol(M, g) = SO(n), we might obtain R(g) = 0 at x0by considering the
element e17→ −e1, (e3, e4)7→ (e4, e3) which is an element in SO(n) to show
that
(3.30) (R1331 +R1441) + (R2332 +R2442) = 0.
32 Man-Chun Lee, Luen-Fai Tam, Jingbo Wan
Since χIC 1≥0, we must have R1331 +R1441 = 0. Using parallel transport with
Hol(M, g) = SO(n) again, we conclude that R(g)≡0 which is impossible.
This proves our claim. Hence in this case, the lemma is true.
Suppose Hol(M, g) = U( n
2). Then by [24], (M, g) is K¨ahler with positive
orthogonal bisectional curvature. It follows from [7, 8, 29] that the normalized
Ricci flow from g(t) converges to CPn/2as t→Tmax after rescaling. Hence the
curvature of g(t) also tends to infinity.
We now apply Theorem 3.2 and Lemma 3.5 to study the rigidity of area
non-increasing maps in the following two cases:
(E): (Nn, h0) and (Mm, g0) satisfy
(3.31)
(Nn, h0) is Einstein with κN≥0;
(Mm, g0) is locally irreducible and non-symmetric with χI C 1(g0)≥0;
Rmin(g0)≥m
nRmax(h0)
(F): (Mm, g0) is locally irreducible and non-symmetric and (Nn, h0) satisfies
(3.32) τN≤0, χIC 1(g0)≥0,and Rmin (g0)≥m
nRmax(h0)
where τNdenotes an upper bound of the sectional curvature of h0and R
denotes the scalar curvature.
Theorem 3.3. Suppose (Mm, g0)and (Nn, h0)be two compact manifolds with
ℓ= min{m, n} ≥ 3such that (E) or (F) holds. If f0is a smooth map from M
to Nwhich is area non-increasing from (M, g0)to (N, h0), then we have the
following.
(i) (Mm, g0)is not Einstein, and f0is homotopically trivial; or
(ii) (Mm, g0)is Einstein, and either f0is homotopical trivial or f0is local
isometry (if m=n), isometric immersion (if m < n).
(iii) If in addition τN<0in case of (F), then f0is homotopy trivial.
Proof. We only prove the case (E) while the case (F) is proved similarly.
(i) Suppose g0is not Einstein. Let us first assume that
Rmin(g0)>m
nRmax(h0).
Then we can shrink h0to a2h0for some 0 < a < 1, so that f0is strictly area
decreasing so that the above inequality on scalar curvatures is still true for
g0, a2h0. Hence without loss of generality, we may assume that f0is strictly
area decreasing. Let g(t), t ∈[0, Tmax) be the maximal solution of the Ricci
flow starting from g0. By Lemma 3.5, Tmax <∞. Since h0is a Einstein metric,
h(t) = (1 −Lt)h0for some L≥0 where Ric(h0) = Lh. Hence h(t) is defined
on [0, L−1).Here L−1is understood to be +∞if L= 0. By considering the
Rigidity of area non-increasing maps 33
lower bound of scalar curvature of g(t), the strong maximum principle and the
fact that g0is not Einstein, we have
(3.33) R(g(t)) >mL
1−Lt =m
nR(h(t))
for t∈(0, Tmax) and in particular Tmax < L−1. By Theorem 3.2, we can
solve the graphical mean curvature flow F:M→(M×N, g(t)⊕h(t)) with
F0= Id ×f0which exists on [0, Tmax). On the other hand, by Lemma 3.5, the
sectional curvature of g(t) tends to infinity as t→Tmax while the sectional
curvature of h(t) remains bounded in [0, Tmax] because Tmax < L−1. This
reduces to the situation in Theorem 3.1, and hence f0is homotopically trivial.
If we only assume that
Rmin(g0)≥m
nRmax(h0),
then (3.33) is still true for t > 0 by strong maximum principle. Let Fbe
the short time solution of the graphical mean curvature flow as above, by
Theorem 3.2, ft: (M, g(t)) →(N, h(t)) is still area non-increasing. Moreover
g(t) is still in χI C1by [3]. Hence ftis homotopically trivial by the above
discussion and hence f0is also homotopically trivial.
(ii) Suppose g0is Einstein. Then the Einstein constant must be positive
because g0is χIC1and is locally irreducible. So g0has positive Ricci curvature
and hence its universal cover is compact. By Lemma 3.5, g0has positive
sectional curvature. If m > n, then by the assumption on the scalar curvatures
of g0, h0
Ric(g0)min −Ric(g0)max + (m−n)κM>0
where κMis the lower bound of the sectional curvature of g0. We can shrink
h0a little so that the above inequality is true and f0is strictly area decreasing.
By Theorem 3.1(ii), we conclude that f0is homotopically trivial.
Since g0has positive sectional curvature, by Theorem 3.1(iii) we conclude
that either f0is homotopically trivial or f0is a local isometry (if m=n) and
isometric immersion (if m < n).
The proof of (iii) is similar, and we omit the details.
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Rigidity of area non-increasing maps 35
(Man-Chun Lee) Department of Mathematics, The Chinese University of Hong
Kong, Shatin, Hong Kong, China
Email address:mclee@math.cuhk.edu.hk
(Luen-Fai Tam) The Institute of Mathematical Sciences and Department of
Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong,
China.
Email address:lftam@math.cuhk.edu.hk
(Jingbo Wan) Department of Mathematics, Columbia University, New York,
NY, 10027
Email address:jingbowan@math.columbia.edu