CURVATURE FLOW IN HYPERBOLIC SPACES
BEN ANDREWS AND XUZHONG CHEN
Abstract. We study the evolution of compact convex hypersurfaces in hyperbolic space
Hn+1 , with normal speed governed by the curvature. We concentrate mostly on the case
of surfaces, and show that under a large class of natural flows, any compact initial surface
with Gauss curvature greater than 1 produces a solution which converges to a point in finite
time, and becomes spherical as the final time is approached. We also consider the higherdimensional case, and show that under the mean curvature flow a similar result holds if the
initial hypersurface is compact with positive Ricci curvature.
1. introduction
In this paper, we consider compact hypersurfaces Mt = Xt (M ) in hyperbolic space Hn+1 that
contract with normal velocity equal to F , according to the evolution equation
∂
(1)
X = −F ν,
∂t
where F is a function of the principal curvatures of the surface, and ν is the outer unit normal
of Mt . There are many papers which consider the evolution of hypersurfaces under flows of this
kind, beginning with the work of Huisken [Hu1] on mean curvature flow, and including flow by
powers of Gauss curvature [T, C1], scalar curvature [C2], and large classes of other examples
[A1, A5]. Several previous papers have considered such flows in the special case of surfaces in
three-dimensional space [A3, A6, S1, S2, S3, L], where the low dimensional setting allows a more
complete understanding of the equation for the evolution of the second fundamental form, and
where there is also a more general regularity theory available [A7]. In particular, in Euclidean
space [A3], compact convex surfaces moving by their Gauss curvature become spherical as they
shrink to points, and the same is true if the speed of motion is an arbitrary monotone increasing
homogeneous degree one function of the principal curvatures [A6]. The authors [AC] recently
proved results of this kind if the speed is a power of the Gauss curvature.
For hypersurfaces in non-Euclidean background spaces, the understanding of the behaviour
of these flows is less complete: The first author [A2]
√ found a flow which takes any compact
hypersurface with principal curvatures greater that c in a Riemannian background space with
sectional curvatures at least −c (with c ≥ 0), and deforms it to a point with spherical limiting
shape. However this flow is rather special and the behaviour of flows such as mean curvature
flow, Gauss curvature flow, and other examples are not well understood in this setting.
In this paper we consider the corresponding questions for surfaces in hyperbolic spaces. The
negative curvature of the background space produces terms which prevent the same estimates
from being applied in the hyperbolic setting, and it is necessary to find different improving
quantities in order to control the geometry of the evolving surfaces. We concentrate mostly on
2010 Mathematics Subject Classification. 53C44, 35K55, 58J35.
The first author was partly supported by Discovery Projects grant DP1200097 of the Australian Research
Council. The second author was partly supported by National Natural Science Foundation of China under grants
11271132, 11071212 and 11131007. The authors are grateful for the hospitality of the Mathematical Sciences
Center of Tsinghua University where the research was carried out.
1
2
BEN ANDREWS AND XUZHONG CHEN
the case of surfaces, for which we study several classes of examples: If F is equal to the mean
curvature H, so that the surfaces move by the mean curvature flow, previous results [CRM]
apply when the principal curvatures κi are bounded below by 1 (in which case one says that
the surface is horospherically convex ). Huisken [Hu2] allowed a weaker condition, that κi H > 2
for each i. Here we show that the condition can be weakened to that of positive intrinsic scalar
curvature R = 2(K − 1): We show that the mean curvature flow preserves this condition, and
evolves any compact surface with R > 0 to a point in finite time, with spherical limiting shape.
We prove a similar result for an analogue of the Gauss curvature flow, in which the speed F
equals K − 1 = R/2: This flow contracts surfaces with positive intrinsic scalar curvature to
round points. Finally, we provide a generalisation of the results of [A6] to the hyperbolic setting:
In [A6] the speeds could be an arbitrary monotone increasing homogeneous degree one function
F̃ of the principal curvatures, but in the hyperbolic case we instead choose the speed F to have
the form F = (1 − 1/K)F̃ . In the final section of the paper we consider the higher dimensional
situation, for which we consider only the case of motion by mean curvature. Previous results
require horospherical convexity, and we show that this can be weakened to positive intrinsic Ricci
curvature. In all of these results the principal difficulty is to find suitable inequalities on the
second fundamental form which are preserved by the evolution
equation.
o
n
p
n+1
We identify H
with the future timelike unit sphere (x, x0 ) ∈ Rn+1 × R : x0 = 1 + |x|2
in Minkowski space Rn+1,1 ' Rn+1 × R, so that we can rescale embeddings into Hn+1 as maps
into the vector space Rn+1,1 . We will always assume that M is compact and connected. Our
main result for the n = 2 case is as follows:
Theorem 1. Let X0 : M 2 → H3 be an embedding with positive scalar curvature R = 2(K − 1).
Then for any smooth, symmetric, function F of principal curvatures with positive derivatives in
each argument, there exists a unique solution X : M × [0, T ) → H3 of (1) with X(z, 0) = X0 (z)
for all z ∈ M , on a maximal time interval [0, T ). Furthermore, if either F = H, F = K − 1 or
F = (1 − 1/K)F̃ , where F̃ is a smooth, homogeneous degree one function of principal curvatures
which has positive derivative with respect to each argument, then Mt = X(M, t) is strictly convex
for each t ∈ [0, T ), and X(., t) converges uniformly to p ∈ H3 as t → T . In these cases the
solutions are asymptotic to a sphrinking sphere as t → T , in the following sense: If Op ∈ O(3, 1)
is the Lorenz boost which brings p to the point e0 ∈ R3,1 , then the rescaled immersions X̃(z, t) =
Op (X(z,t))−e0
converge in C ∞ to a limiting immersion X̃T with image equal to the unit sphere
r(t)
in R3 ⊂ R3,1 , with kX̃t − X̃T kC k ≤ Ck (T − t)a for each k for some a > 0. Here
q we can choose
p
1/3
r(t) = 4(T − t) for F = H, r(t) = (3(T − t))
for F = K − 1, and r(t) = 2F̃ (1, 1)(T − t)
for F = (1 − 1/K)F̃ .
The key step in the proof is a pinching estimate for the principal curvatures, analogous to
those used in [A3] and [A6]. The details of the precise pinching estimate, and the subsequent
analysis, differ substantially in the three cases, so we present the proofs of each case separately.
Our result for hypersurfaces in higher dimensional hyperbolic spaces is as follows:
Theorem 2. For any embedding X0 : M n → Hn+1 with positive Ricci curvature, there exists
a smooth solution of the mean curvature flow (equation (1) with F = H) on a maximal time
interval [0, T ). The hypersurfaces Mt = Xt (M ) have positive Ricci curvature for each t ∈ (0, T ),
and are asymptotic to a sphrinking sphere as t → T , in the following sense: If Op ∈ O(n+1, 1) is
the Lorenz boost which brings p to the point e0 ∈ Rn+1,1 , then the rescaled immersions X̃(z, t) =
Op (X(z,t))−e0
√
converge in C ∞ to a limiting immersion X̃T with image equal to the unit sphere in
2n(T −t)
Rn+1 ⊂ Rn+1,1 , with kX̃t − X̃T kC k ≤ Ck (T − t)a for each k for some a > 0.
CURVATURE FLOW IN HYPERBOLIC SPACES
3
We remark that restricting to hypersurfaces in Hn+1 does not exclude consideration of surfaces
in other hyperbolic manifolds: The assumption of positive Ricci curvature guarantees that M has
compact universal cover, so any immersion of M into a hyperbolic manifold lifts to an immersion
of the universal cover into Hn+1 , and the evolution equation (1) respects this lifting. Thus our
results also hold in this more general setting.
2. notation and preliminary results
Throughout the paper we adopt the Einstein summation convention of summing over repeated
indices from 1 to n unless otherwise specified.
We denote by u·v the Minkowski inner product of two vectors in Rn+1,1 , defined by e0 ·e0 = −1
and ei · ei = 1 for i = 1, . . . , n + 1, with ei · ej for i 6= j. We consider immersions X : M →
Hn+1 ⊂ Rn+1,1 in which the induced inner product on T M defined by g(u, v) = uX · vX has
positive intrinsic Ricci curvature (here uX denotes the derivative of X in direction u). At each
point we have a normal line within TX(x) Hn+1 defined by Nx M = {n ∈ Rn+1,1 : n · uX =
0 for all u ∈ Tx M ; n · X = 0}. For each unit vector ν ∈ Nx M we have a second fundamental
form hν : Tx M × Tx M → R defined by hν (u, v) = −uvX · ν. The principal curvatures κi
are the eigenvalues of hν with respect to the inner product g, and the principal directions are
the corresponding eigenvectors; the mean curvature is H is the sum of the principal curvatures,
and we denote by |A|2 the sum of the shares of the principal curvatures. The scalar curvature
is given by the Gauss equation as R = H 2 − |A|2 − n(n − 1), so the positive Ricci curvature
assumption implies that |H| > n everywhere. We can therefore choose the unit normal so that
H is positive. Henceforward we make this choice of ν, but omit the subscript ν and refer to the
second fundamental form simply by h. The Ricci curvature in the direction of the ith principal
direction is given by κi (H − κi ) − (n − 1), so each κi is positive and the hypersurface is locally
convex. Given a basis {∂i } for Tx M we define the components of g and h in this basis by
gij = g(∂i , ∂j ) and hij = h(∂i , ∂j ). The cotangent space T ∗ M then has dual basis {dxi } defined
by dxi (∂j ) = δji , and the dual metric on T ∗ M then has components g ij given by the inverse of
the matrix gij .
The Weingarten relation provides a formula for the derivative of the unit normal vector ν:
uν · vX = h(u, v),
(2)
or equivalently, with respect to a basis,
∂i ν = hik g kl ∂l X = hli ∂l X.
The second derivatives of the immersion X can be decomposed as follows:
(3)
uvX = −νh(u, v) + (∇u v)X − Xg(u, v),
where ∇ is the Levi-Civita connection of the metric g. The Codazzi equation then gives
∇u h(v, w) = ∇v h(u, w)
(4)
for all u, v, w ∈ Tx M . Combining the Gauss and Codazzi equations gives the following generalisation of Simons’ identity (see for example [A2]):
(5)
∇(i ∇j) hkl = ∇(k ∇l) hij + hij hpk hpl − hkl hpi hpj − gij hkl + gkl hij ,
where the brackets denote symmetrisation.
Also note that at a given point p ∈ M we can always choose coordinates so that
gij = δij
∇
∂
∂xi
∂
=0
∂xj
hij = diag(κ1 , . . . , κn ).
4
BEN ANDREWS AND XUZHONG CHEN
The normal velocity F can be considered as a function of (κ1 , κ2 ) or (hij , gij ). We set
2
F
∂F
, F̈ ij, kl = ∂h∂ij ∂h
. Note that in an orthonormal frame with hij = diag(κ1 , . . . , κn )
Ḟ = ∂h
ij
kl
ij
∂F
we also have Ḟ ij = diag(Ḟ 1 , . . . , Ḟ n ) where Ḟ i = ∂κ
.
i
Under the evolution equation (1) we have the following evolution equations (see [A2]):
(6)
(7)
(8)
(9)
∂
gij = −2F hij ,
∂t
∂
F = Ḟ ij ∇i ∇j F + F Ḟ ij hip hp j − F Ḟ ij gij ,
∂t
∂
hij = Ḟ kl ∇k ∇l hij + F̈ kl,mn ∇i hkl ∇j hmn − Ḟ kl hkl hpi hpj + Ḟ kl hij hpk hpl
∂t
− Ḟ kl hkl gij + Ḟ kl gkl hij − F hki hkj − F gij ,
∂
G = Ḟ ij ∇i ∇j G + Ġij F̈ kl,mn − Ḟ ij G̈kl,mn ∇i hkl ∇j hmn
∂t
− Ġij Ḟ kl hkl hpi hpj + Ġij Ḟ kl hij hpk hpl − Ġij Ḟ kl gij hkl
+ Ġij Ḟ kl gkl hij + F Ġij hki hkj − F Ġij gij .
where G = G(hij ) is a symmetric function of the principal curvatures. In the special case n = 2,
F = K − 1 this becomes
∂
gij = − 2(K − 1)hij ,
∂t
∂
(11)
H =K̇ ij ∇i ∇j H + g ij K̈ ij,kl ∇i hkl ∇j hmn + 2(K − 1)2 ,
∂t
∂
(12)
K =K̇ ij ∇i ∇j K + (K − 1)2 H.
∂t
Finally, for F = H, we have the following:
(10)
∂
gij = − 2Hhij ,
∂t
∂
2
(14)
hij =∆hij − 2H(hik hkj + gij ) + (|A| + n)hij ,
∂t
∂
2
(15)
H =∆H + (|A| − n)H.
∂t
If F is a smooth function of the principal curvatures defined on some open subset A of Rn ,
which is strictly increasing in each argument, and X0 is a smooth immersion of a compact nmanifold as a hypersurface in Hn+1 such that the principal curvatures at each point lie in A, then
there exists a unique smooth solution for a short time interval (see for example the treatment
given in [B]). We define T to be the maximal interval of existence of a solution. In the remainder
of the paper our argument will show that for the particular classes of speed F we consider,
the evolving hypersurfaces contract to points and become spherical in shape at the end of the
maximal interval of existence.
We conclude this section by observing that the maximal time of existence is necessarily finite
for a very large class of flows:
(13)
Proposition 1. Suppose F is defined and positive on a domain A which contains (c, . . . , c) for
c > 1, and is nondecreasing (in the sense that if (a1 , . . . , an ) and (b1 , . . . , bn ) are in A with
bi ≥ ai for all i, then F (b1 , . . . , bn ) ≥ F (a1 , . . . , an )), then the maximal time of existence for any
compact solution of (1) is finite.
CURVATURE FLOW IN HYPERBOLIC SPACES
5
Proof. Let d be the hyperbolic distance from any fixed point in H3 . Then d is smooth where it
is nonzero, and we have the following evolution equation:
∂d
= Dd(−F ν).
∂t
At a point where the spatial maximum of d is attained, we have
0 = ∇i d = Dd(∂i ),
so that ν points in the radial direction, and
0 ≥ ∇j ∇i d = D2 d(∂i , ∂j ) − Dd(hij ν).
But since ν is radial we have
D2 d(∂i , ∂j ) = coth dgij ,
so the second derivative condition becomes
κi ≥ coth d
for all i. Since F is an increasing function of the principal curvatures, this implies that F ≥
F (coth d, . . . , coth d) > 0. The maximum principle (in the form given in [Ha1, Lemma 3.5])
implies that the maximum of d is non-increasing, so we have d ≤ d0 = supx∈M d(X(x, 0)).
The monotonicity of F and the fact that coth is a decreasing function then gives F ≥ F0 =
F (coth d0 , . . . , coth d0 ), and so
∂
d ≤ −F0 ,
∂t
so that supx∈M d(X(x, t)) ≤ d0 − F0 t. Since d is manifestly non-negative, we conclude that
T ≤ d0 /F0 .
3. scalar curvature flow
In this section, we will study the K − 1 flow for n = 2:
∂
X = −(K − 1)ν
∂t
where K is Gauss curvature and ν is the outer normal vector of Mt . In this case the crucial
estimate is a bound on the difference between the principal curvatures, which follows a similar
argument to that in [A3].
(16)
3.1. Pinching estimate. We first prove that positive scalar curvature of the initial surface is
preserved by equation (16):
Proposition 2. If K(x, 0) > 1 for all x ∈ M , then K(x, t) > 1 for all x ∈ M and all t ∈ [0, T )
under (16). As a consequence, the surfaces Mt are strictly convex for all t ∈ [0, T ).
Proof. This follows from the evolution equation (12) for Gauss curvature K, and maximum
principle.
Next we prove the bound on the difference between the principal curvatures:
Theorem 3. Let {Mt = X(M, t)}0≤t≤T be a smooth connected solution of the flow equation
(16) with K > 1. Then
(17)
sup |κ1 (p, t) − κ2 (p, t)| ≤ sup |κ1 (p, 0) − κ2 (p, 0)|
p∈M
p∈M
6
BEN ANDREWS AND XUZHONG CHEN
Proof. We will apply the maximum principle to the quantity
Q = H 2 − 4K = (κ1 − κ2 )2
We first compute an evolution equation for Q:
∂H
∂K
∂Q
=2H
− 4K
∂t
∂t
∂t
ij
=2H(K̇ ∇i ∇j H + g ij K̈ ij,kl ∇i hkl ∇j hmn + 2(K − 1)2 )
−4(K̇ ij ∇i ∇j K + (K − 1)2 H)
=K̇ ij ∇i ∇j Q − 2K̇ ij ∇i H∇j H + 2Hg ij K̈(∇i h, ∇j h)
Suppose p is a point in M where a maximum of Q is attained at time t ∈ [0, T ), choose local
coordinates for M near p such that gij = δij and hij is diagonal. At this point the leading term
on the right-hand side of the above evolution equation is non-positive. We now estimate the
remaining terms. Using the fact that ∇Q = 0 at p
0 =∇1 Q = 2H∇1 H − 4∇1 K = 2(κ1 + κ2 )(∇1 h11 + ∇2 h22 )
−4κ1 ∇1 h22 − 4κ2 ∇1 h11
=2(κ1 − κ2 )(∇1 h11 − ∇1 h22 )
If κ1 = κ2 , then Q = 0 and we have nothing to prove. So we can assume that ∇1 h11 = ∇1 h22 at
the point p. Similarly we have ∇2 h11 = ∇2 h22 . Now we compute
K̈(∇1 h, ∇1 h) =2∇1 h11 ∇1 h22 − 2(∇1 h12 )2
=2(∇1 h11 )2 − 2(∇2 h11 )2
=2(∇1 h11 )2 − 2(∇2 h22 )2
By using ∇Q = 0 condition and the Codazzi identity. Similarly
K̈(∇2 h, ∇2 h) = 2(∇2 h22 )2 − 2(∇1 h11 )2
therefore at the point p, we have
g ij K̈(∇i h, ∇j h) = K̈(∇1 h, ∇1 h) + K̈(∇2 h, ∇2 h)
Thus the last term on the right-hand side of the evolution equation for Q vanishes and the second term is manifestly non-positive. The maximum principle applies to show that the supremum
of Q over M is non-increasing function of time.
Corollary 4. Under the conditions of Theorem 3 there exists C1 such that 0 <
1
C1
≤
κ2
κ1
≤ C1 .
Proof. Theorem 3 and the inequality K > 1 imply that
κ1
κ2
(κ1 − κ2 )2
+
−2=
≤ (κ1 − κ2 )2 ≤ C0
κ2
κ1
κ1 κ2
where C0 ≡ supp∈M (κ1 (p, 0) − κ2 (p, 0))2 .
Corollary 5. Under the conditions of Theorem 3 there exists C2 such that κi ≥ C2 for i = 1, 2.
Proof. This is a direct consequence of Corollary 4 and the inequality K > 1.
CURVATURE FLOW IN HYPERBOLIC SPACES
7
3.2. Convergence. In this section we discuss the convergence of the solution to a point and of
the rescaled immersions to the sphere. The argument differs from that in previous work [A2, S2]
in only a few points. As in other flows with speed growing super linearly in the curvature, the
main difficulty is in the non-uniform parabolicity of the flow. This difficulty can be overcome
using either the methods of [S3, AS, WTL] for which the key step is to apply estimates for porous
medium equations, or the methods of [AM] which use geometric estimates to show the surface
becomes close to a sphere near the final time, and then a barrier argument to force the speed
to become positive so that the flow becomes uniformly parabolic. We sketch here the former
approach.
We first observe that the Gauss curvature must become unbounded as the final time is approached: If not, then since we have a positive lower bound on K − 1 by Proposition 2, and a
bound on ratios of principal curvatures by Corollary 4, all principal curvatures are bounded above
and below by positive constants. In particular the coefficients Ḟ ij have eigenvalues bounded
above and below by positive constants. The results of [A7] (applied in a local graph parametrisation, for example) apply to give C 0,α bounds on the second fundamental form, and higher
derivative estimates follow by standard Schauder estimates. It follows that the solution can be
extended further, following an argument similar to that in [Hu1].
We therefore consider a sequence of times tk approaching T , for which a new maximum of the
Gauss curvature is attained, so that
(18)
λ2k =
sup
K(x, t) = K(xk , tk )
x∈M,0≤t≤tk
for some xk ∈ M for each k. Now let Ok ∈ O(3, 1) be such that Ok (X(xk , tk ) = e0 and
ν(xk , tk ) = e3 , and define
(19)
Xk (x, t) = λk Ok X(x, tk + λ−3
k t) − e0 .
Then Xk : M × [−λ3k tk , λ3k (T − tk )) → λk (H3 − e0 ) is a solution of the equation
∂Xk
= −(Kk − λ−2
k )νk .
∂t
where Kk is the Gauss curvature of the solution Xk and νk is the unit normal (within λk (H3 −e0 )).
Furthermore we have supx∈M,t≤0 Kk (x, t) = Kk (xk , 0) = 1 for each x, νk (M, 0)) = e3 and
Xk (xk , 0) = 0.
The result of Corollary 4 implies that ratios of principal curvatures are bounded at each point
for Xk , and hence that all principal curvatures are bounded since K ≤ 1.
Under equation (20) the Gauss curvature evolves as follows:
(20)
∂
2
K = K̇ ij ∇i ∇j K + (K − λ−2
k ) H.
∂t
The evolving surfaces may be written locally as graphs over a totally geodesic hyperbolic 2-plane
H in H3 : At each point x ∈ H let ν(x) be the unit normal. Then we define Ψ : H × R → H3 by
Ψ(x, s) = expx (sν(x)), and describe Mt locally by the embedding
(21)
(22)
X̃ : x 7→ Ψ(x, u(x, t)).
The inner product induced by Ψ is given by
(23)
g(Ψ∗ (u, a), Ψ∗ (v, b)) = cosh2 (λ−1
k s)ḡ(u, v) + ab,
where ḡ is the hyperbolic metric on H. It follows that the tangent vectors to the embedding are
given by
(24)
∂i X = ∂i Ψ + ui ∂s Ψ,
8
BEN ANDREWS AND XUZHONG CHEN
where ui = ∂i u, so that the induced inner product is
gij = cosh2 (λ−1
k u)ḡij + ui uj
(25)
and the unit normal to the surface is given by the expression
(26)
−ui ḡ ij ∂j Ψ + cosh2 (λ−1
k u)∂s Ψ
q
.
2 −1
2
cosh(λ−1
cosh
(λ
u)
u)
+
|Du|
ḡ
k
k
ν=
The evolution of the graphical embedding is related to X by a time-dependent diffeomorphism
defined by the requirement that the orthogonal projection onto H remains fixed. Explicitly, this
is given by
∂ X̃
k
= −(K − λ−2
k )ν − V ∂k X,
∂t
(27)
where
(28)
Vk =
(K − λ−2
)up ḡ pk
q k
.
2 −1
2
cosh(λ−1
cosh
(λ
u)
u)
+
|Du|
ḡ
k
k
It follows that the evolution of K under the graphical flow is governed by the following equation:
∂K
2
k
= K̇ ij ∇i ∇j K + (K − λ−2
(29)
k ) H − V ∇k K.
∂t
The Codazzi identity implies that ∇i K̇ ij = 0, so we also have the following divergence form:
∂K
2
k
= ∇i K̇ ij ∇j K + (K − λ−2
k ) H − V ∇k K
∂t
¯ i K̇ ij ∇
¯ j K + (Γ − Γ̄)ip p K̇ pj − V j ∇
¯ j K + (K − λ−2 )2 H,
=∇
(30)
k
where Γij k and Γ̄ij k are the Christoffel symbols of the Levi-Civita connections of g and ḡ on H,
respectively. These are related by the Levi-Civita formula:
1
¯ i gjq + ∇
¯ j giq − ∇
¯ q gij ,
(31)
Γij k − Γ̄ij k = g kq ∇
2
¯ p log det g.
which implies that (Γ − Γ̄)ip i = 12 ∇
The bound on curvature, together with barrier arguments as in [A2, Lemma 5.2], gives the
following: There exists r > 0 such that for each k and each x ∈ M , the hypersurfaces can be
written as hyperbolic graphs of the form (22) on a region Gr = {(z, t) : z ∈ Br (x), −r2 ≤ t ≤ 0},
¯
¯ 2 u(z, t)|ḡ ≤ r−1 for all (z, t) ∈ Gr . From
with 0 ≤ u(z, t) ≤ r−1 , |∇u(z,
t)|ḡ ≤ r−1 , and |∇
equation (30) we deduce that on each of these solutions the Gauss curvature evolves according
to an equation of the form
¯ i aij ∇
¯ j K 3/2 + Ai ∇
¯ iK + f
(32)
Kt = ∇
where λδ ij ≤ aij ≤ Λδ ij for some 0 < λ < Λ, and Ai and f are bounded. This follows since
ij
K̇ ij K −1/2 = K 1/2 h−1
has positive bounds above and below by Corollary 4.
Theorem 1.2 of [DF] can now be applied to give a Hölder continuity estimate for K on Gr/2 ,
R
¯ 3/2 |2 dµ(ḡ). To bound the latter we observe that (since K̇ ij
with constant depending on Gr |∇K
is comparable to K 1/2 g ij ),
Z
Z
5
d
2
K 5/2 ≤ −
K 3/2 ∇i K̇ ij ∇j K + (K − λ−2
)
H
+ K 5/2 H(K − λ−2
k
k )dµ
dt M
2 M
Z
≤ −C
|∇i K 3/2 |2 + C,
M
CURVATURE FLOW IN HYPERBOLIC SPACES
so that (since g and ḡ are comparable on Gr )
Z
Z
Z
3/2 2
3/2 2
¯
|∇K |ḡ dµ(ḡ) ≤ C
|∇K |g dµ(g) ≤ −C
Gr
Mt
t=0
t=−r 2
d
dt
9
Z
K
5/2
dµ(g) + C
dt
Mt
Integrating this gives the required bound.
This proves that K is Hölder continuous on M × [−r2 /2, 0], with constants independent of k.
Therefore in this region for any (x0 , t0 ) we have K comparable to K(x0 , t0 ) on Br (x0 )×[t0 −r2 , t0 ]
where r2 is comparable to K(x0 , t0 ). On this set the evolution equation is uniformly parabolic,
and the estimates of [A7] apply to give Hölder continuity of second derivatives, with estimates
depending on K(x0 , t0 ). Schauder estimates then give C k,α estimates for every k. This proves
that the hypersurfaces Mt have all derivatives of second fundamental form bounded on regions
where K > 0.
It now follows that Xk converges as k → ∞ to a solution of Gauss curvature flow with bounded
curvature in R3 , which is smooth at points where K > 0. The pinching estimate of Theorem 3
implies that |κ2 − κ1 | ≤ Cκ−1
k → 0, so the limiting hypersurface is totally umbilic and hence a
sphere. The smooth (rather than subsequential) convergence follows without difficulty.
4. mean curvature flow (n = 2)
In this section, we will study mean curvature flow in three-dimensional hyperbolic space:
∂
X = −Hν
∂t
where H is mean curvature and ν is outer normal vector of Mt .
(33)
4.1. Pinching estimates. We will prove that scalar curvature of solutions to equation (33)
remains positive if initially so, and also that the principal curvatures remain bounded in ratio.
According to evolution equation (14), we have
∂
2
hij = ∆hij − 2H(hik hkj + gij ) + (|A| + 2)hij
∂t
If we introduce the canonical spacetime connection (as in [AH, Section 6.3] or [AB, Section 2.3])
by setting ∇t ∂i = −Hhji ∂j , then this becomes
(34)
2
∇t hij = ∆hij − 2Hgij + (|A| + 2)hij
We apply the vector bundle maximum principle to the evolution equation (34) to prove that the
inequality K > 1 is preserved by mean curvature flow.
Proposition 3. If K > 1 at t = 0, then K > 1 for all t ∈ [0, T ). In particular, Mt remains
strictly convex.
Proof. We will use the vector bundle maximum principle introduced in [Ha1] (see in particular
the formulation in [AH, Theorem 7.15]). Let κ1 and κ2 be the principal curvatures of Mt . Let
2
Qij = (|A| + 2)hij − 2Hgij , which is the reaction term of (34). In particular, if we work in a
frame where (hij ) is diagonal, then so is Q:
3
κ1 0
κ1 + κ1 κ22 − 2κ2
0
(35)
h=
=⇒ Q =
.
0 κ2
0
κ32 + κ2 κ21 − 2κ1
Consider the following convex domain:
(36)
Ω = {(κ1 , κ2 ) : κ1 κ2 − 1 ≥ 0, κ1 + κ2 > 0}
10
BEN ANDREWS AND XUZHONG CHEN
For the vector bundle maximum principle to apply, we need the vector field Q to point into
Ω, which is equivalent to the derivative of the defining function κ1 κ2 − 1 in direction Q being
positive. We have
Qκi = κi (|A|2 + 2) − 2H,
and so
Q(κ1 κ2 ) = κ1 κ2 (|A|2 + 2) − 2Hκ2 + κ1 κ2 (|A|2 + 2) − 2Hκ1
= 2(κ1 κ2 − 1)|A|2 + 2|A|2 + 4κ1 κ2 − 2H 2
= 2(κ1 κ2 − 2)|A|2
= 0.
Therefore the maximum principle applies, and the Proposition is proved.
Theorem 6. Let {Mt = X(M, t)}0≤t<T be a smooth, strictly convex solution of the flow equation
(33) with K > 1. Then
(37)
sup
p∈M
|κ2 (p, 0) − κ22 (p, 0)|
|κ21 (p, t) − κ22 (p, t)|
≤ sup 1
κ1 (p, t)κ2 (p, t) − 1 p∈M κ1 (p, 0)κ2 (p, 0) − 1
Proof. Let C3 (M0 ) = supp∈M
|κ21 (p,0)−κ22 (p,0)|
κ1 (p,0)κ2 (p,0)−1 .
We need to prove
|κ21 − κ22 | ≤ C3 (κ1 κ2 − 1).
(38)
We again apply the vector bundle maximum principle. To apply this we must verify that the
inequality (38) defines a subset of the bundle of symmetric 2-tensors which is convex in the fibre
and invariant under parallel transport, and such that the vector field Q defined by (35) points
into the set. We verify this using arguments similar to [AH, Section 7.5.3.1–7.5.3.2]: First we
show that this defines a convex subset Ω of the vector bundle of symmetric 2-tensors: Define
det h − 1
Ω = h > 0, h(e1 , e1 ) − h(e2 , e2 ) ≤ C3
for all {e1 , e2 } orthonormal .
trace h
det h
1
Since trace
are concave functions on the positive cone, the function h(e1 , e1 ) −
h and − trace
h
det h−1
h(e2 , e2 ) − C3 trace h is convex for each orthonormal frame {e1 , e2 }. It follows that Ω is an
intersection of convex sets, hence convex. The invariance under parallel transport is automatic
from the definition. It remains to check that the vector field Q points
into Ω. Let h ∈ ∂Ω,
h−1
so that for some frame we have h(e1 , e1 ) − h(e2 , e2 ) − C3 det
.
Variation
of the frame
trace h
shows that e1 and e2 are eigenvectors of h, with corresponding eigenvalues κ1 > κ2 satisfying
κ21 − κ22 = C3 (κ1 κ2 − 1). The supporting linear function at this point is given by
`(A) = [2κ1 − Cκ2 ] A11 + [−2κ2 − Cκ1 ] A22 .
We have Q11 =
κ31
− 2κ2 and Q22 = κ32 + κ2 κ21 − 2κ1 , so this becomes
`(Q) = 2κ21 (κ21 + κ22 ) − 4κ1 κ2 − Cκ2 κ2 κ21 + κ22 + 2Cκ22
+
κ2 κ22
− 2κ22 (κ21 + κ22 ) + 4κ1 κ2 − Cκ2 κ2 (κ21 + κ22 ) + 2Cκ21
= 2(κ21 + κ22 )(κ21 − κ22 − C(κ1 κ2 − 1))
= 0.
The maximum principle therefore applies, and the inequality is preserved by the flow.
Corollary 7. For a smooth compact strictly convex solution of mean curvature flow with K > 1
in H3 , there exists C4 such that 0 < C14 ≤ κκ21 ≤ C4 .
CURVATURE FLOW IN HYPERBOLIC SPACES
Proof. Theorem 6 and K > 1 implies that
2 2
κ2
(κ1 − κ2 )2 (κ1 + κ2 )2
(κ1 − κ2 )2 (κ1 + κ2 )2
κ1
+
−2=
≤
≤ C32 .
κ2
κ1
(κ1 κ2 )2
(κ1 κ2 − 1)2
q
C2
C2
The corollary follows with C42 = 1 + 23 + C3 1 + 43 .
11
4.2. Convergence. The argument for convergence for the mean curvature flow is considerably
simpler that that for the scalar curvature flow of the previous section. We rescale in a similar
way to the previous section, choosing a sequence of times tk on which the mean curvature reaches
a mew maximum, so that
(39)
λk =
sup
H(x, t) = H(xk , tk )
x∈M,0≤t≤tk
for some xk ∈ M for each k. Now let Ok ∈ O(3, 1) be such that Ok (X(xk , tk ) = e0 and
ν(xk , tk ) = e3 , and define
(40)
Xk (x, t) = λk Ok X(x, tk + λ−2
k t) − e0 .
Then Xk : M × [−λ2k tk , λ2k (T − tk )) → λk (H3 − e0 ) is a solution of the mean curvature flow.
Furthermore we have supx∈M,t≤0 Hk (x, t) = Hk (xk , 0) = 1 for each x, νk (M, 0)) = e3 and
Xk (xk , 0) = 0.
Standard estimates (see for example [EH, Theorem 3.4] for the Euclidean mean curvature
flow) yield bounds on all higher derivatives of second fundamental form, independent of k. It
follows that the solutions Xk converge to a complete strictly convex solution of mean curvature
flow in Euclidean space, satisfying the pinching ratio bound of Corollary 7. The result of [Ha2]
implies that the limiting solution consists of compact convex hypersurfaces, and the fact that
the pinching ratio of a compact convex solution of mean curvature flow in Euclidean space is
strictly decreasing unless the hypersurface is a sphere implies that the limiting is a shrinking
sphere solution. The convergence result follows.
5. F -flow
In this section, we will study the flow in three-dimensional hyperbolic space:
∂
X = −F ν
∂t
(41)
where F = (1 −
1
K )F̃
and ν is the outer normal vector of Mt .
5.1. Pinching estimates. The crucial estimate in this case is a bound on the ratio of principal
curvatures.
Theorem 8. Let {Mt = X(M, t)0≤t<T } be a smooth, solution of the flow equation (41) with
K > 1. Then
(42)
sup
M
(κ1 (x, t) − κ2 (x, t))2
(κ1 (x, 0) − κ2 (x, 0))2
≤ sup
2
2
(κ1 (x, t) + κ2 (x, t))
M (κ1 (x, 0) + κ2 (x, 0))
Proof. We will apply maximum principle to the quantity
G=
(κ1 − κ2 )2
(κ1 + κ2 )2
12
BEN ANDREWS AND XUZHONG CHEN
G is a homogeneous degree zero function, so (9) applies to give the following evolution equation:
∂
G =Ḟ ij ∇i ∇j G + Ġij F̈ kl,mn − Ḟ ij G̈kl,mn ∇i hkl ∇j hmn
∂t
(43)
− Ġij Ḟ kl hkl hpi hpj − Ġij Ḟ kl gij hkl + F Ġij hki hkj − F Ġij gij .
Suppose p is a point in M where a new maximum of G is attained at time t ∈ [0, T ). Choose
local normal coordinates for M near p such that hij (p, t) = diag(κ1 , κ2 ). We now estimate the
second term on the right hand side of the above evolution equation. The computation of this
term will require results from [A5] which describe the components of F and G in the above
orthonormal frame:
∂2F
,
∂κ21
∂2F
=
,
∂κ1 ∂κ2
∂2F
= 2,
∂κ2
F̈ 11,11 =
F̈ 11,22 = F̈ 22,11
F̈ 22,22
∂F
∂F
− ∂κ
2
F̈ 12,12 = F̈ 21,21 = ∂κ1
κ1 − κ2
The last of these identities is to be interpreted as a limit if κ1 = κ2 . It follows that the second
term on the right hand side of the evolution equation for G are as follows:
Q = Ġij F̈ kl,mn − Ḟ ij G̈kl,mn ∇i hkl ∇j hmn
∂G ∂ 2 F
∂G ∂ 2 F
∂F ∂ 2 G
∂F ∂ 2 G
2
=
−
(∇1 h11 ) +
−
(∇1 h22 )2
∂κ1 ∂κ21
∂κ1 ∂κ21
∂κ1 ∂κ22
∂κ1 ∂κ22
∂F ∂ 2 G
∂G ∂ 2 F
−
∇1 h11 ∇1 h22
+2
∂κ1 ∂κ1 ∂κ2
∂κ1 ∂κ1 ∂κ2
∂G ∂ 2 F
∂G ∂ 2 F
∂F ∂ 2 G
∂F ∂ 2 G
2
−
(∇2 h11 ) +
−
(∇2 h22 )2
+
∂κ2 ∂κ21
∂κ2 ∂κ21
∂κ2 ∂κ22
∂κ2 ∂κ22
∂G ∂ 2 F
∂F ∂ 2 G
+2
−
∇2 h11 ∇2 h22
∂κ2 ∂κ1 ∂κ2
∂κ2 ∂κ1 ∂κ2
+2
∂G ∂F
∂κ1 ∂κ2
−
∂G ∂F
∂κ2 ∂κ1
κ2 − κ1
2
(∇1 h12 ) + 2
∂G ∂F
∂κ1 ∂κ2
−
∂G ∂F
∂κ2 ∂κ1
κ2 − κ1
(∇2 h12 )2 .
1
F̃
F̃
Now we note F = (1 − K
)F̃ = F̃ − K
. Let S = K
, so that
Q = Ġij F̈ kl,mn − Ḟ ij G̈kl,mn ∇i hkl ∇j hmn
= Ġij F̃¨ kl,mn − F̃˙ ij G̈kl,mn ∇i hkl ∇j hmn − Ġij S̈ kl,mn − Ṡ ij G̈kl,mn ∇i hkl ∇j hmn
=QF̃ − QS ,
where
QS = Ġij S̈ kl,mn − Ṡ ij G̈kl,mn ∇i hkl ∇j hmn .
At a maximum point (p, t) of G, G is non-zero (otherwise Mt is a sphere and the proof is
trivial) and it can be assumed without loss of generality κ1 > κ2 because the maximum point is
CURVATURE FLOW IN HYPERBOLIC SPACES
13
not umbilic. The gradient conditions on G then give two equations:
∇1 h11 = −
∂G
∂κ2
∂G
∂κ1
∇1 h22 ,
∇2 h22 = −
∂G
∂κ1
∂G
∂κ2
∇2 h11
∂G
∂G
+ κ2 ∂κ
= 0.
The degree-zero homogeneity of G implies by the Euler relation that κ1 ∂κ
1
2
Homogeneity also implies the identity
κ21
∂2G
∂2G
∂2G
+ κ22 2 + 2κ1 κ2
=0
2
∂κ1
∂κ2
∂κ1 ∂κ2
Similarly, the degree 1 homogeneity of F̃ and the degree −1 homogeneity of S give the following
identities
κ2 ∂ 2 F̃
∂ 2 F̃
=
−
;
∂κ21
κ1 ∂κ1 ∂κ2
∂ 2 F̃
κ1 ∂ 2 F̃
=
−
;
∂κ22
κ2 ∂κ1 ∂κ2
κ1
∂ F̃
∂ F̃
+ κ2
= F̃
∂κ1
∂κ2
and
κ21
2
∂2S
∂2S
2∂ S
+
2κ
κ
= 2S;
+
κ
1
2
2
∂κ21
∂κ1 κ2
∂κ22
κ1
∂S
∂S
+ κ2
= −S
∂κ1
∂κ2
Substituting these expressions into QF̃ and QS and applying the Codazzi symmetries ∇1 h12 =
∇2 h11 and ∇2 h12 = ∇1 h22 , we obtain
QF̃ =
∂G
2F̃ ∂κ
1
κ2 (κ2 − κ1 )
(∇1 h22 )2 +
∂G
2F̃ ∂κ
1
κ2 (κ2 − κ1 )
(∇2 h11 )2
and
QS =
∂G
∂κ1
2S
2S
−
κ22
κ2 (κ2 − κ1 )
(∇1 h22 )2 +
∂G
∂κ2
2S
2S
−
κ21
κ1 (κ2 − κ1 )
(∇2 h11 )2
2
1 −κ2 )
The derivatives of the function G = (κ
(κ1 +κ2 )2 with respect to κ1 and κ2 can be computed as
follows:
∂G
4κ2 (κ1 − κ2 )
∂G
4κ1 (κ2 − κ1 )
(44)
=
;
=
∂κ1
(κ1 + κ2 )3
∂κ2
(κ1 + κ2 )3
This gives the following:
QF̃ = −
QS =
8F̃
(∇1 h22 )2 + (∇2 h11 )2
3
(κ1 + κ2 )
8F̃
8F̃
(∇1 h22 )2 +
(∇2 h11 )2
(κ1 + κ2 )3 κ22
(κ1 + κ2 )3 κ21
In particular QF̃ is non-positive and QS is non-negative, so Q = QF̃ − QS ≤ 0. Now we consider
the terms on the second line of (43):
Z = − Ġij Ḟ kl hkl hpi hpj − Ġij Ḟ kl gij hkl + F Ġij hki hkj − F Ġij gij
=Ġij hpi hpj (−Ḟ kl hkl + F ) − Ġij gij (Ḟ kl hkl + F )
∂G 2
∂G 2
∂F
∂F
∂G
∂G
∂F
∂F
=
κ1 +
κ2
−
κ1 −
κ2 + F −
+
κ1 +
κ2 + F
∂κ1
∂κ2
∂κ1
∂κ2
∂κ1
∂κ2
∂κ1
∂κ2
The derivatives of the function F = F̃ (1 −
(45)
∂F
∂ F̃
1
F̃ κ2
=
(1 − ) + 2 ;
∂κ1
∂κ1
K
K
1
K)
with respect to κ1 and κ2 are as follows:
∂F
∂ F̃
1
F̃ κ1
=
(1 − ) + 2
∂κ2
∂κ2
K
K
14
BEN ANDREWS AND XUZHONG CHEN
Substituting the expression (44) and (45) into the expression for Z , we can obtain
Z=−
4(κ1 − κ2 )2
2F̃ 4κ1 κ2 (κ1 − κ2 )2
+
2
F̃
=0
K
(κ1 + κ2 )3
(κ1 + κ2 )3
Therefore by the maximum principle, the maximum of G is non-increasing.
Corollary 9. For a smooth compact strictly convex surface Mt in H3 , flowing according to
κ2
∂X
1
∂t = −F ν, there exists C6 = C6 (M0 ) such that 0 < C6 ≤ κ1 ≤ C6 .
Proof. This is a direct consequence of Theorem 8.
5.2. Uniform parabolicity. We next deduce that the evolution equation (41) is uniformly
parabolic: Since F = (1 − 1/K)F̃ , we have
F̃
Ḟ ij = (1 − 1/K)F̃˙ ij + 2 K̇ ij .
K
The set C = {A : |A| = 1,
1
C6
≤
κ2 (A)
κ1 (A)
≤ C6 } is compact, so the eigenvalues of F̃˙ ij attain a
finite maximum and a positive minimum value on this set. Since F̃ is homogeneous of degree
one, we have F̃˙ ij (h) = F̃˙ ij (h/|h|). Since h/|h| ∈ C, we have C− g ij ≤ F̃˙ ij ≤ C+ g ij for some
0 < C− ≤ C+ .
Similarly, by homogeneity and the pinching ratio bound, we have that the eigenvalues of
F̃
ij
are in an interval of the form [C̃− /K, C̃+ /K] for some 0 < C̃− ≤ C̃+ . This gives the
K 2 K̇
bounds
#
"
#
"
(1 − 1/K)C− +
C̃− ij
C̃+ ij
g ≤ Ḟ ij ≤ (1 − 1/K)C+ +
g .
K
K
Since K > 1 the bracket on the right hand side is bounded and the bracket on the left is bounded
away from zero.
5.3. Speed bounds and positive scalar curvature. The next important result is a lower
bound on the speed: From the equation (7) and the estimtes of the previous section, we have
∂
F ≥ Ḟ ij ∇i ∇j F − CF
∂t
for some C. Therefore inf x∈M F (x, t) ≥ e−Ct inf x∈M F (x, 0), and since the maximal time of
existence is finite, the speed F has a positive lower bound on the entire interval of existence.
It follows also that K −1 has a positive lower bound throughout the evolution, since (K −1) =
K
F
. By the pinching ratio bound we have K
≥ C F̃ ≥ CF for some C, so K − 1 ≥ CF 2 has a
F̃
F̃
positive lower bound.
5.4. Convergence. The argument for convergence is now similar to that given for mean curvature flow above: First, the regularity estimates of [A7], together with Schauder estimates,
apply to show that the solution continues to exist as long as the speed remains bounded. We
can therefore rescale at a sequences of times approaching the final time on which F attains new
spatial maximum values approaching infinity, and product a limit which is a solution of the flow
∂X
∂t = −F̃ ν in Euclidean space, with ratio of principal curvatures bounded. The result of [Ha2]
implies that the hypersurfaces of the limiting solution are compact, and the monotonicity of the
pinching ratio implies that the limit solution is a shrinking sphere.
CURVATURE FLOW IN HYPERBOLIC SPACES
15
6. Mean curvature flow (n > 2)
In this section we consider the higher-dimensional situation of convex hypersurfaces with
positive Ricci curvature in the hyperbolic space Hn+1 . We restrict our attention to the case of
the mean curvature flow (F = H).
6.1. Preserving positive Ricci curvature. We begin by showing that the condition of positive Ricci curvature is preserved under the mean curvature flow. The positive Ricci curvature
condition can be written as follows (since the normal direction was chosen to make all principal
curvatures positive):
n
n \
\
n−1
>0 .
Ω = {Ric > 0} =
{κi > 0} ∩
H − κi −
κi
i=1
i=1
Each of the sets in this intersection is the super-level set of a concave function, and hence is a
convex set. Therefore Ω is a symmetric convex subset of Rn . It follows that the set of bilinear
forms representing second fundamental forms giving positive Ricci curvature is an O(n)-invariant
convex subset of Sym2 (Rn ). To apply the vector bundle maximum principle as in [AH, Section
7.5.3], it remains only to prove that the vector field Q representing the reaction terms in equation
(34) points into Ω at any boundary point. That is, we must show that at a boundary point where
one of the defining functions fi = κi (H −κi )−(n−1) vanishes, the derivative Qfi of fi in direction
Q is non-negative. The kth component of Q at the point (κ1 , . . . , κn ) is
Q(κk ) = κk (|A|2 + n) − 2H.
(46)
Therefore we have
X
Qfi = κi (|A|2 + n) − 2H (H − κi ) + κi
κj (|A|2 + n) − 2H
j6=i
= 2κi (H − κi )(|A|2 + n) − 2H(H − κi ) − 2(n − 1)κi H
= 2 (κi (H − κi ) − (n − 1)) |A|2 + 2(n − 1)|A|2 + 2nκi (H − κi ) − 2(H − κi )2
− 2κi (H − κi ) − 2(n − 1)κi (H − κi ) − 2(n − 1)κ2i

2
X
X
= 2 (κi (H − κi ) − (n − 1)) |A|2 + 2(n − 1)
κ2j − 
κj 
j6=i
= 2fi |A|2 +
X
j6=i
2
(κj − κk )
j,k6=i
(47)
2
≥ 2fi |A| .
In particular, since fi = 0 we have Qfi ≥ 0 and the maximum principle therefore applies to
prove that the principal curvature remain in Ω, and the condition of positive Ricci curvature is
preserved under the mean curvature flow.
6.2. The pinching estimate. The crucial step in controlling the hypersurfaces is to obtain an
estimate which bounds the ratio of principal curvatures under the mean curvature flow, for any
initial hypersurface with positive Ricci curvature. The precise result is as follows:
Proposition 4. For any compact solution of the mean curvature flow with positive Ricci curvature, there exists ε > 0 such that the following inequality holds at every point (x, t) ∈ M × [0, T ):
(48)
for all i, j and k.
κk (H − κk ) − (n − 1) ≥ εH(κi − κj ),
16
BEN ANDREWS AND XUZHONG CHEN
Proof. We will again employ the vector bundle maximum principle, noting that by compactness
and the positivity of the Ricci curvature the inequality holds for sufficiently small ε > 0 when
t = 0. We must first show that the inequality defines a convex set. Dividing the inequality (48)
by H, we find that the set we are trying to preserve can be written in the form
\
\ n−1
κ2k
−
− εκi + εκj ≥ 0 .
Ωε = {κi > 0} ∩
κk −
H
H
i
i,j,k
The function in the last bracket is concave: The terms κk − εκj + εκi are linear, and −1/H is
clearly concave. The function −κ2k /H is also concave, since it is homogeneous of degree one and
manifestly concave on the hyperplane {H = 1}. Therefore Ωε is an intersection of superlevel sets
of concave functions, hence is a convex set.
It remains only to prove that the vector field Q points into Ωε at boundary points. Equivalently, we must show that at a boundary point (κ1 , . . . , κn ) where one of the defining functions
fi,j,k = κk (H − κk ) − (n − 1) − εH(κi − κj ) is zero, the derivative Qfi,j,k is non-negative, where
Q is given by (46). We note that fi,j,k = fk − εφi,j where fk is defined as in section 6.1 and
φi,j = H(κi − κj ). Since Qκi = κi (|A|2 + n) − 2H we have
QH = H(|A|2 + n) − 2nH = H(|A|2 − n).
Also we have
Q(κi − κj ) = κi (|A|2 + n) − 2H − κj (|A|2 + n) + 2H = (κi − κj )(|A|2 + n).
Combining these gives
Qφi,j = (QH)(κi − κj ) + HQ(κi − κj )
= H(κi − κj )(|A|2 − n) + H(κi − κj )(|A|2 + n)
= 2H(κi − κj )|A|2
= 2gi,j |A|2 .
Combining this with equation 47 gives
Qfi,j,k = Qfk − εQgi,j ≥ 2fk |A|2 − 2εgi,j |A|2 = 2fi,j,k |A|2 = 0.
The maximum principle therefore applies, and we have proved (48).
Corollary 10. Under the assumptions of Proposition 4 there exists C > 0 such that
κi (x, t)
≤C
C −1 ≤
κj (x, t)
for all x ∈ M and t ∈ [0, T ).
Proof. Labelling the principal curvatures in increasing order, we have
1
1
1
H(κn − κ1 ) ≤ (κ1 (H − κ1 ) − (n − 1)) ≤ κ1 (H − κ1 ) ≤ κ1 H.
ε
ε
ε
Dividing through by H we find
1
κn − κ1 ≤ κ1 ,
ε
and therefore
1
κn ≤ 1 +
κ1 ,
ε
so the corollary holds with C = 1 + 1ε .
6.3. Rescaling and convergence. The argument for convergence given in Section 4.2 applies
for this case without change.
CURVATURE FLOW IN HYPERBOLIC SPACES
17
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Mathematical Sciences Center, Tsinghua University; and Mathematical Sciences Institute, Australia National University.
E-mail address: [email protected]
Department of Mathematics, East China Normal University
E-mail address: [email protected]