CURVATURE FLOW IN HYPERBOLIC SPACES
BEN ANDREWS AND XUZHONG CHEN
Abstract. We study the evolution of compact convex hypersurfaces in hyperbolic space n+1 H , 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 higher- dimensional 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 n+1 In this paper, we consider compact hypersurfaces Mt = Xt(M) in hyperbolic space H 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 κiH > 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 F˜ of the principal curvatures, but in the hyperbolic case we instead choose the speed F to have the form F = (1 − 1/K)F˜. 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. n o n+1 n+1 p 2 We identify H with the future timelike unit sphere (x, x0) ∈ R × R : x0 = 1 + |x| 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:
2 3 Theorem 1. Let X0 : M → H 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 3 each argument, there exists a unique solution X : M × [0,T ) → H 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)F˜, where F˜ 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) 3,1 ˜ is the Lorenz boost which brings p to the point e0 ∈ R , then the rescaled immersions X(z, t) = Op(X(z,t))−e0 ∞ ˜ r(t) converge in C to a limiting immersion XT with image equal to the unit sphere 3 3,1 ˜ ˜ a in R ⊂ R , with kXt − XT kCk ≤ Ck(T − t) for each k for some a > 0. Here we can choose q r(t) = p4(T − t) for F = H, r(t) = (3(T − t))1/3 for F = K − 1, and r(t) = 2F˜(1, 1)(T − t) for F = (1 − 1/K)F˜. 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:
n n+1 Theorem 2. For any embedding X0 : M → H 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 n+1,1 ˜ the Lorenz boost which brings p to the point e0 ∈ R , then the rescaled immersions X(z, t) = Op(X(z,t))−e0 ∞ √ converge in C to a limiting immersion X˜T with image equal to the unit sphere in 2n(T −t) n+1 n+1,1 ˜ ˜ a R ⊂ R , with kXt − XT kCk ≤ Ck(T − t) 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. n+1,1 We denote by u·v the Minkowski inner product of two vectors in R , 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 TM defined by g(u, v) = uX · vX has positive intrinsic Ricci curvature (here uX denotes the derivative of X in direction u). At each n+1 n+1,1 point we have a normal line within TX(x)H defined by NxM = {n ∈ R : n · uX = 0 for all u ∈ TxM; n · X = 0}. For each unit vector ν ∈ NxM we have a second fundamental form hν : TxM × TxM → 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 = H2 − |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 TxM we define the components of g and h in this basis by ∗ i gij = g(∂i, ∂j) and hij = h(∂i, ∂j). The cotangent space T M then has dual basis {dx } defined i i ∗ ij by dx (∂j) = δj, and the dual metric on T M then has components g given by the inverse of the matrix gij. The Weingarten relation provides a formula for the derivative of the unit normal vector ν: (2) uν · vX = h(u, v), or equivalently, with respect to a basis, kl l ∂iν = hikg ∂lX = hi∂lX. The second derivatives of the immersion X can be decomposed as follows:
(3) uvX = −νh(u, v) + (∇uv)X − Xg(u, v), where ∇ is the Levi-Civita connection of the metric g. The Codazzi equation then gives
(4) ∇uh(v, w) = ∇vh(u, w) for all u, v, w ∈ TxM. Combining the Gauss and Codazzi equations gives the following generali- sation of Simons’ identity (see for example [A2]): p p (5) ∇(i∇j)hkl = ∇(k∇l)hij + hijhkhpl − hklhi hpj − gijhkl + gklhij, where the brackets denote symmetrisation. Also note that at a given point p ∈ M we can always choose coordinates so that
∂ i gij = δij ∇ ∂ = 0 hj = diag(κ1, . . . , κn). ∂xi ∂xj 4 BEN ANDREWS AND XUZHONG CHEN
The normal velocity F can be considered as a function of (κ1, κ2) or (hij, gij). We set ij ∂F ij, kl ∂2F F˙ = , F¨ = . Note that in an orthonormal frame with hij = diag(κ1, . . . , κn) ∂hij ∂hij ∂hkl we also have F˙ ij = diag(F˙ 1,..., F˙ n) where F˙ i = ∂F . ∂κi Under the evolution equation (1) we have the following evolution equations (see [A2]): ∂ (6) g = −2F h , ∂t ij ij ∂ (7) F = F˙ ij∇ ∇ F + F F˙ ijh hp − F F˙ ijg , ∂t i j ip j ij ∂ (8) h = F˙ kl∇ ∇ h + F¨kl,mn∇ h ∇ h − F˙ klh hph + F˙ klh hph ∂t ij k l ij i kl j mn kl i pj ij k pl ˙ kl ˙ kl k − F hklgij + F gklhij − F hi hkj − F gij, ∂ (9) G = F˙ ij∇ ∇ G + G˙ ijF¨kl,mn − F˙ ijG¨kl,mn ∇ h ∇ h ∂t i j i kl j mn ˙ ij ˙ kl p ˙ ij ˙ kl p ˙ ij ˙ kl − G F hklhi hpj + G F hijhkhpl − G F gijhkl ˙ ij ˙ kl ˙ ij k ˙ ij + G F gklhij + F G hi hkj − F G gij.
i where G = G(hj) is a symmetric function of the principal curvatures. In the special case n = 2, F = K − 1 this becomes ∂ (10) g = − 2(K − 1)h , ∂t ij ij ∂ (11) H =K˙ ij∇ ∇ H + gijK¨ ij,kl∇ h ∇ h + 2(K − 1)2, ∂t i j i kl j mn ∂ (12) K =K˙ ij∇ ∇ K + (K − 1)2H. ∂t i j Finally, for F = H, we have the following: ∂ (13) g = − 2Hh , ∂t ij ij ∂ (14) h =∆h − 2H(h hk + g ) + (|A|2 + n)h , ∂t ij ij ik j ij ij ∂ (15) H =∆H + (|A|2 − 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 n- manifold 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: 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 = ∇id = Dd(∂i), so that ν points in the radial direction, and 2 0 ≥ ∇j∇id = D d(∂i, ∂j) − Dd(hijν). But since ν is radial we have 2 D 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 ≤ −F , ∂t 0 so that supx∈M d(X(x, t)) ≤ d0 − F0t. 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: ∂ (16) 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].
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
2 2 Q = H − 4K = (κ1 − κ2) We first compute an evolution equation for Q: ∂Q ∂H ∂K =2H − 4K ∂t ∂t ∂t ij ij ij,kl 2 =2H(K˙ ∇i∇jH + g K¨ ∇ihkl∇jhmn + 2(K − 1) ) ij 2 −4(K˙ ∇i∇jK + (K − 1) H) ij ij ij =K˙ ∇i∇jQ − 2K˙ ∇iH∇jH + 2Hg K¨ (∇ih, ∇jh) 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 =∇1Q = 2H∇1H − 4∇1K = 2(κ1 + κ2)(∇1h11 + ∇2h22)
−4κ1∇1h22 − 4κ2∇1h11
=2(κ1 − κ2)(∇1h11 − ∇1h22)
If κ1 = κ2, then Q = 0 and we have nothing to prove. So we can assume that ∇1h11 = ∇1h22 at the point p. Similarly we have ∇2h11 = ∇2h22. Now we compute
2 K¨ (∇1h, ∇1h) =2∇1h11∇1h22 − 2(∇1h12) 2 2 =2(∇1h11) − 2(∇2h11) 2 2 =2(∇1h11) − 2(∇2h22) By using ∇Q = 0 condition and the Codazzi identity. Similarly
2 2 K¨ (∇2h, ∇2h) = 2(∇2h22) − 2(∇1h11) therefore at the point p, we have
ij g K¨ (∇ih, ∇jh) = K¨ (∇1h, ∇1h) + K¨ (∇2h, ∇2h) Thus the last term on the right-hand side of the evolution equation for Q vanishes and the sec- ond term is manifestly non-positive. The maximum principle applies to show that the supremum of Q over M is non-increasing function of time.
1 κ2 Corollary 4. Under the conditions of Theorem 3 there exists C1 such that 0 < ≤ ≤ C1. C1 κ1 Proof. Theorem 3 and the inequality K > 1 imply that
2 κ1 κ2 (κ1 − κ2) 2 + − 2 = ≤ (κ1 − κ2) ≤ C0 κ2 κ1 κ1κ2 2 where C0 ≡ supp∈M (κ1(p, 0) − κ2(p, 0)) .
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 ap- proached: 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 F˙ ij have eigenvalues bounded above and below by positive constants. The results of [A7] (applied in a local graph parametri- sation, for example) apply to give C0,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 2 (18) λk = 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 −3 (19) Xk(x, t) = λk OkX(x, tk + λk t) − e0 . 3 3 3 Then Xk : M × [−λktk, λk(T − tk)) → λk(H − e0) is a solution of the equation ∂X (20) k = −(K − λ−2)ν . ∂t k k k 3 where Kk is the Gauss curvature of the solution Xk and νk is the unit normal (within λk(H −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: ∂ (21) K = K˙ ij∇ ∇ K + (K − λ−2)2H. ∂t i j k 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 (22) X˜ : x 7→ Ψ(x, u(x, t)). The inner product induced by Ψ is given by 2 −1 (23) g(Ψ∗(u, a), Ψ∗(v, b)) = cosh (λk s)¯g(u, v) + ab, whereg ¯ is the hyperbolic metric on H. It follows that the tangent vectors to the embedding are given by
(24) ∂iX = ∂iΨ + ui∂sΨ, 8 BEN ANDREWS AND XUZHONG CHEN where ui = ∂iu, so that the induced inner product is 2 −1 (25) gij = cosh (λk u)¯gij + uiuj and the unit normal to the surface is given by the expression ij 2 −1 −uig¯ ∂jΨ + cosh (λk u)∂sΨ (26) ν = q . −1 2 −1 2 cosh(λk u) cosh (λk u) + |Du|g¯ 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 ∂X˜ (27) = −(K − λ−2)ν − V k∂ X, ∂t k k where −2 pk k (K − λk )upg¯ (28) V = q . −1 2 −1 2 cosh(λk u) cosh (λk u) + |Du|g¯ It follows that the evolution of K under the graphical flow is governed by the following equation: ∂K (29) = K˙ ij∇ ∇ K + (K − λ−2)2H − V k∇ K. ∂t i j k k ij The Codazzi identity implies that ∇iK˙ = 0, so we also have the following divergence form: ∂K = ∇ K˙ ij∇ K + (K − λ−2)2H − V k∇ K ∂t i j k k ¯ ˙ ij ¯ ¯ p ˙ pj j ¯ −2 2 (30) = ∇i K ∇jK + (Γ − Γ)ip K − V ∇jK + (K − λk ) H, k k where Γij and Γ¯ij are the Christoffel symbols of the Levi-Civita connections of g andg ¯ on H, respectively. These are related by the Levi-Civita formula: 1 (31) Γ k − Γ¯ k = gkq ∇¯ g + ∇¯ g − ∇¯ g , ij ij 2 i jq j iq q ij ¯ i 1 ¯ which implies that (Γ − Γ)ip = 2 ∇p log det g. 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 2 written as hyperbolic graphs of the form (22) on a region Gr = {(z, t): z ∈ Br(x), −r ≤ t ≤ 0}, −1 −1 2 −1 with 0 ≤ u(z, t) ≤ r , |∇¯ u(z, t)|g¯ ≤ r , and |∇¯ u(z, t)|g¯ ≤ r for all (z, t) ∈ Gr. From equation (30) we deduce that on each of these solutions the Gauss curvature evolves according to an equation of the form ij 3/2 i (32) Kt = ∇¯ i a ∇¯ jK + A ∇¯ iK + f where λδij ≤ aij ≤ Λδij for some 0 < λ < Λ, and Ai and f are bounded. This follows since ij K˙ ijK−1/2 = K1/2 h−1 has positive bounds above and below by Corollary 4. Theorem 1.2 of [DF] can now be applied to give a H¨oldercontinuity estimate for K on Gr/2, with constant depending on R |∇¯ K3/2|2dµ(¯g). To bound the latter we observe that (since K˙ ij Gr is comparable to K1/2gij), Z Z d 5/2 5 3/2 ˙ ij −2 2 5/2 −2 K ≤ − K ∇iK ∇jK + (K − λk ) H + K H(K − λk )dµ dt M 2 M Z 3/2 2 ≤ −C |∇iK | + C, M CURVATURE FLOW IN HYPERBOLIC SPACES 9 so that (since g andg ¯ are comparable on Gr) Z Z Z t=0 Z ¯ 3/2 2 3/2 2 d 5/2 |∇K |g¯dµ(¯g) ≤ C |∇K |gdµ(g) ≤ −C K dµ(g) + C dt 2 Gr Mt t=−r dt Mt Integrating this gives the required bound. This proves that K is H¨older continuous on M × [−r2/2, 0], with constants independent of k. 2 Therefore in this region for any (x0, t0) we have K comparable to K(x0, t0) on Br(x0)×[t0 −r , t0] 2 where r is comparable to K(x0, t0). On this set the evolution equation is uniformly parabolic, and the estimates of [A7] apply to give H¨oldercontinuity of second derivatives, with estimates k,α depending on K(x0, t0). Schauder estimates then give C 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 −1 implies that |κ2 − κ1| ≤ Cκk → 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: ∂ (33) X = −Hν ∂t where H is mean curvature and ν is outer normal vector of Mt.
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 ∂ h = ∆h − 2H(h hk + g ) + (|A|2 + 2)h ∂t ij ij ik j ij ij If we introduce the canonical spacetime connection (as in [AH, Section 6.3] or [AB, Section 2.3]) j by setting ∇t∂i = −Hhi ∂j, then this becomes 2 (34) ∇thij = ∆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 2 κ1 0 κ1 + κ1κ2 − 2κ2 0 (35) h = =⇒ Q = 3 2 . 0 κ2 0 κ2 + κ2κ1 − 2κ1 Consider the following convex domain:
(36) Ω = {(κ1, κ2): κ1κ2 − 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 κ1κ2 − 1 in direction Q being positive. We have 2 Qκi = κi(|A| + 2) − 2H, and so 2 2 Q(κ1κ2) = κ1κ2(|A| + 2) − 2Hκ2 + κ1κ2(|A| + 2) − 2Hκ1 2 2 2 = 2(κ1κ2 − 1)|A| + 2|A| + 4κ1κ2 − 2H 2 = 2(κ1κ2 − 2)|A| = 0.
Therefore the maximum principle applies, and the Proposition is proved.
Theorem 6. Let {Mt = X(M, t)}0≤t
2 2 |κ1(p,0)−κ2(p,0)| Proof. Let C3(M0) = sup . We need to prove p∈M κ1(p,0)κ2(p,0)−1 2 2 (38) |κ1 − κ2| ≤ C3(κ1κ2 − 1). 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(e , e ) − h(e , e ) ≤ C for all {e , e } orthonormal . 1 1 2 2 3 trace h 1 2 det h 1 Since trace h and − trace h are concave functions on the positive cone, the function h(e1, e1) − 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, det h−1 so that for some frame we have h(e1, e1) − h(e2, e2) − C3 trace h . Variation of the frame shows that e1 and e2 are eigenvectors of h, with corresponding eigenvalues κ1 > κ2 satisfying 2 2 κ1 − κ2 = C3(κ1κ2 − 1). The supporting linear function at this point is given by
`(A) = [2κ1 − Cκ2] A11 + [−2κ2 − Cκ1] A22. 3 2 3 2 We have Q11 = κ1 + κ2κ2 − 2κ2 and Q22 = κ2 + κ2κ1 − 2κ1, so this becomes 2 2 2 2 2 2 `(Q) = 2κ1(κ1 + κ2) − 4κ1κ2 − Cκ2κ2 κ1 + κ2 + 2Cκ2 2 2 2 2 2 2 − 2κ2(κ1 + κ2) + 4κ1κ2 − Cκ2κ2(κ1 + κ2) + 2Cκ1 2 2 2 2 = 2(κ1 + κ2)(κ1 − κ2 − C(κ1κ2 − 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 3 1 κ2 in , there exists C4 such that 0 < ≤ ≤ C4. H C4 κ1 CURVATURE FLOW IN HYPERBOLIC SPACES 11
Proof. Theorem 6 and K > 1 implies that
2 2 2 2 2 2 κ1 κ2 (κ1 − κ2) (κ1 + κ2) (κ1 − κ2) (κ1 + κ2) 2 + − 2 = 2 ≤ 2 ≤ C3 . κ2 κ1 (κ1κ2) (κ1κ2 − 1)
2 q 2 2 C3 C3 The corollary follows with C4 = 1 + 2 + C3 1 + 4 .
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 −2 (40) Xk(x, t) = λk OkX(x, tk + λk t) − e0 . 2 2 3 Then Xk : M × [−λktk, λk(T − tk)) → λk(H − 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: ∂ (41) X = −F ν ∂t 1 ˜ where F = (1 − K )F 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
2 (κ1 − κ2) G = 2 (κ1 + κ2) 12 BEN ANDREWS AND XUZHONG CHEN
G is a homogeneous degree zero function, so (9) applies to give the following evolution equation:
∂ ij ij kl,mn ij kl,mn G =F˙ ∇i∇jG + G˙ F¨ − F˙ G¨ ∇ihkl∇jhmn (43) ∂t ˙ ij ˙ kl p ˙ ij ˙ kl ˙ ij k ˙ ij − G F hklhi hpj − G F gijhkl + F G hi hkj − F G 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: 2 ¨11,11 ∂ F F = 2 , ∂κ1 ∂2F F¨11,22 = F¨22,11 = , ∂κ1∂κ2 2 ¨22,22 ∂ F F = 2 , ∂κ2 ∂F − ∂F F¨12,12 = F¨21,21 = ∂κ1 ∂κ2 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:
ij kl,mn ij kl,mn Q = G˙ F¨ − F˙ G¨ ∇ihkl∇jhmn 2 2 2 2 ∂G ∂ F ∂F ∂ G 2 ∂G ∂ F ∂F ∂ G 2 = 2 − 2 (∇1h11) + 2 − 2 (∇1h22) ∂κ1 ∂κ1 ∂κ1 ∂κ1 ∂κ1 ∂κ2 ∂κ1 ∂κ2 ∂G ∂2F ∂F ∂2G +2 − ∇1h11∇1h22 ∂κ1 ∂κ1∂κ2 ∂κ1 ∂κ1∂κ2 2 2 2 2 ∂G ∂ F ∂F ∂ G 2 ∂G ∂ F ∂F ∂ G 2 + 2 − 2 (∇2h11) + 2 − 2 (∇2h22) ∂κ2 ∂κ1 ∂κ2 ∂κ1 ∂κ2 ∂κ2 ∂κ2 ∂κ2 ∂G ∂2F ∂F ∂2G +2 − ∇2h11∇2h22 ∂κ2 ∂κ1∂κ2 ∂κ2 ∂κ1∂κ2 ∂G ∂F − ∂G ∂F ∂G ∂F − ∂G ∂F ∂κ1 ∂κ2 ∂κ2 ∂κ1 2 ∂κ1 ∂κ2 ∂κ2 ∂κ1 2 +2 (∇1h12) + 2 (∇2h12) . κ2 − κ1 κ2 − κ1 1 ˜ ˜ F˜ F˜ Now we note F = (1 − K )F = F − K . Let S = K , so that
ij kl,mn ij kl,mn Q = G˙ F¨ − F˙ G¨ ∇ihkl∇jhmn
ij ¨kl,mn ˙ ij kl,mn ij kl,mn ij kl,mn = G˙ F˜ − F˜ G¨ ∇ihkl∇jhmn − G˙ S¨ − S˙ G¨ ∇ihkl∇jhmn
=QF˜ − QS, where ij kl,mn ij kl,mn QS = G˙ S¨ − S˙ G¨ ∇ihkl∇jhmn.
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: ∂G ∂G ∂κ2 ∂κ1 ∇1h11 = − ∂G ∇1h22, ∇2h22 = − ∂G ∇2h11 ∂κ1 ∂κ2 ∂G ∂G The degree-zero homogeneity of G implies by the Euler relation that κ1 + κ2 = 0. ∂κ1 ∂κ2 Homogeneity also implies the identity 2 2 2 2 ∂ G 2 ∂ G ∂ G κ1 2 + κ2 2 + 2κ1κ2 = 0 ∂κ1 ∂κ2 ∂κ1∂κ2 Similarly, the degree 1 homogeneity of F˜ and the degree −1 homogeneity of S give the following identities 2 ˜ 2 ˜ 2 ˜ 2 ˜ ˜ ˜ ∂ F κ2 ∂ F ∂ F κ1 ∂ F ∂F ∂F ˜ 2 = − ; 2 = − ; κ1 + κ2 = F ∂κ1 κ1 ∂κ1∂κ2 ∂κ2 κ2 ∂κ1∂κ2 ∂κ1 ∂κ2 and 2 2 2 2 ∂ S ∂ S 2 ∂ S ∂S ∂S κ1 2 + 2κ1κ2 + κ2 2 = 2S; κ1 + κ2 = −S ∂κ1 ∂κ1κ2 ∂κ2 ∂κ1 ∂κ2
Substituting these expressions into QF˜ and QS and applying the Codazzi symmetries ∇1h12 = ∇2h11 and ∇2h12 = ∇1h22, we obtain 2F˜ ∂G 2F˜ ∂G ∂κ1 2 ∂κ1 2 QF˜ = (∇1h22) + (∇2h11) κ2(κ2 − κ1) κ2(κ2 − κ1) and ∂G 2S 2S 2 ∂G 2S 2S 2 QS = 2 − (∇1h22) + 2 − (∇2h11) ∂κ1 κ2 κ2(κ2 − κ1) ∂κ2 κ1 κ1(κ2 − κ1) 2 (κ1−κ2) The derivatives of the function G = 2 with respect to κ1 and κ2 can be computed as (κ1+κ2) follows:
∂G 4κ2(κ1 − κ2) ∂G 4κ1(κ2 − κ1) (44) = 3 ; = 3 ∂κ1 (κ1 + κ2) ∂κ2 (κ1 + κ2) This gives the following: ˜ 8F 2 2 QF˜ = − 3 (∇1h22) + (∇2h11) (κ1 + κ2) ˜ ˜ 8F 2 8F 2 QS = 3 2 (∇1h22) + 3 2 (∇2h11) (κ1 + κ2) κ2 (κ1 + κ2) κ1
In particular QF˜ is non-positive and QS is non-negative, so Q = QF˜ − QS ≤ 0. Now we consider the terms on the second line of (43): ˙ ij ˙ kl p ˙ ij ˙ kl ˙ ij k ˙ ij Z = − G F hklhi hpj − G F gijhkl + F G hi hkj − F G gij ˙ ij p ˙ kl ˙ ij ˙ kl =G hi hpj(−F hkl + F ) − G gij(F 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 ˜ 1 The derivatives of the function F = F (1 − K ) with respect to κ1 and κ2 are as follows:
∂F ∂F˜ 1 F˜ κ2 ∂F ∂F˜ 1 F˜ κ1 (45) = (1 − ) + 2 ; = (1 − ) + 2 ∂κ1 ∂κ1 K K ∂κ2 ∂κ2 K K 14 BEN ANDREWS AND XUZHONG CHEN
Substituting the expression (44) and (45) into the expression for Z , we can obtain
˜ 2 2 2F 4κ1κ2(κ1 − κ2) ˜ 4(κ1 − κ2) Z = − 3 + 2F 3 = 0 K (κ1 + κ2) (κ1 + κ2) Therefore by the maximum principle, the maximum of G is non-increasing.
3 Corollary 9. For a smooth compact strictly convex surface Mt in H , flowing according to ∂X 1 κ2 = −F ν, there exists C6 = C6(M0) such that 0 < ≤ ≤ C6. ∂t C6 κ1 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)F˜, we have
F˜ F˙ ij = (1 − 1/K)F˜˙ ij + K˙ ij. K2
1 κ2(A) ˙ ij The set C = {A : |A| = 1, ≤ ≤ C6} is compact, so the eigenvalues of F˜ attain a C6 κ1(A) finite maximum and a positive minimum value on this set. Since F˜ is homogeneous of degree ˙ ij ˙ ij ij ˙ ij ij one, we have F˜ (h) = F˜ (h/|h|). Since h/|h| ∈ C, we have C−g ≤ F˜ ≤ C+g for some 0 < C− ≤ C+. Similarly, by homogeneity and the pinching ratio bound, we have that the eigenvalues of F˜ ˙ ij ˜ ˜ ˜ ˜ K2 K are in an interval of the form [C−/K, C+/K] for some 0 < C− ≤ C+. This gives the bounds " # " # C˜ C˜ (1 − 1/K)C + − gij ≤ F˙ ij ≤ (1 − 1/K)C + + gij. − 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 ≥ F˙ ij∇ ∇ F − CF ∂t i j −Ct for some C. Therefore infx∈M F (x, t) ≥ e infx∈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 ≥ CF˜ ≥ CF for some C, so K − 1 ≥ CF 2 has a F˜ F˜ positive lower bound.
5.4. Convergence. The argument for convergence is now similar to that given for mean cur- vature 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 = −F 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 pos- itive 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 Ω = {Ric > 0} = {κ > 0} ∩ H − κ − > 0 . i i κ i=1 i=1 i 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 n convex subset of Sym2(R ). 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 2 (46) Q(κk) = κk(|A| + n) − 2H. Therefore we have 2 X 2 Qfi = κi(|A| + n) − 2H (H − κi) + κi κj(|A| + n) − 2H j6=i 2 = 2κi(H − κi)(|A| + n) − 2H(H − κi) − 2(n − 1)κiH 2 2 2 = 2 (κi(H − κi) − (n − 1)) |A| + 2(n − 1)|A| + 2nκi(H − κi) − 2(H − κi) 2 − 2κi(H − κi) − 2(n − 1)κi(H − κi) − 2(n − 1)κi 2 2 X 2 X = 2 (κi(H − κi) − (n − 1)) |A| + 2(n − 1) κj − κj j6=i j6=i 2 X 2 = 2fi|A| + (κj − κk) j,k6=i 2 (47) ≥ 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 curva- ture, there exists ε > 0 such that the following inequality holds at every point (x, t) ∈ M × [0,T ):
(48) κk(H − κk) − (n − 1) ≥ εH(κi − κj), for all i, j and k. 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 \ \ κ2 n − 1 Ω = {κ > 0} ∩ κ − k − − εκ + εκ ≥ 0 . ε i k H H i j i i,j,k
The function in the last bracket is concave: The terms κk − εκj + εκi are linear, and −1/H is 2 clearly concave. The function −κk/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. Equiva- lently, 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 2 φi,j = H(κi − κj). Since Qκi = κi(|A| + n) − 2H we have QH = H(|A|2 + n) − 2nH = H(|A|2 − n). Also we have 2 2 2 Q(κi − κj) = κi(|A| + n) − 2H − κj(|A| + n) + 2H = (κi − κj)(|A| + n). Combining these gives
Qφi,j = (QH)(κi − κj) + HQ(κi − κj) 2 2 = H(κi − κj)(|A| − n) + H(κi − κj)(|A| + n) 2 = 2H(κi − κj)|A| 2 = 2gi,j|A| . Combining this with equation 47 gives 2 2 2 Qfi,j,k = Qfk − εQgi,j ≥ 2fk|A| − 2εgi,j|A| = 2fi,j,k|A| = 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 κ (x, t) C−1 ≤ i ≤ C 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(κ − κ ) ≤ (κ (H − κ ) − (n − 1)) ≤ κ (H − κ ) ≤ κ H. n 1 ε 1 1 ε 1 1 ε 1 Dividing through by H we find 1 κ − κ ≤ κ , n 1 ε 1 and therefore 1 κ ≤ 1 + κ , n ε 1 1 so the corollary holds with C = 1 + ε . 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
References [AS] Roberta Alessandroni and Carlo Sinestrari, Evolution of hypersurfaces by powers of the scalar curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 3, 541–571. [A1] Ben Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151–171. [A2] , Contraction of convex hypersurfaces in Riemannian spaces, J. Differential Geom. 39 (1994), no. 2, 407–431. [A3] , Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151–161. [A4] B. Andrews, Positively curved surfaces in the Three-sphere, ICM 2002 III (2002), no. 1-3, 221–230. [A5] Ben Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math. 608 (2007), 17–33. [A6] , Moving surfaces by non-concave curvature functions, Calc. Var. Partial Differential Equations 39 (2010), no. 3-4, 649–657. [A7] , Fully nonlinear parabolic equations in two space variables, available at http://arxiv.org/abs/ math/0402235. [AB] Ben Andrews and Charles Baker, Mean curvature flow of pinched submanifolds to spheres, J. Differential Geom. 85 (2010), no. 3, 357–395. [AC] Ben Andrews and Xuzhong Chen, Surfaces moving by powers of Gauss curvature, Pure Appl. Math. Q. 8 (2012), no. 4, 825–834. [AH] Ben Andrews and Christopher Hopper, The Ricci flow in Riemannian geometry, Lecture Notes in Math- ematics, vol. 2011, Springer, Heidelberg, 2011. [AM] Ben Andrews and James McCoy, Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3427–3447. [B] Charles Baker, The mean curvature flow of submanifolds of high codimension, PhD Thesis, The Australian National University, 2010, arXiv:1104.4409[math.DG]. [CRM] Esther Cabezas-Rivas and Vicente Miquel, Volume preserving mean curvature flow in hyperbolic space, preprint, 22 pages, available at arXiv:math/0611216v1[math.DG]. [CN] Q.-M. Cheng and H. Nagakawa, Totally umbilic hypersurfaces, Hiroshima Math. J. 20 (1990), 1–10. [C1] Bennett Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117–138. [C2] , Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), no. 1, 63–82. [DF] Emmanuele DiBenedetto and Avner Friedman, H¨older estimates for nonlinear degenerate parabolic sys- tems, J. Reine Angew. Math. 357 (1985), 1–22. [EH] Klaus Ecker and Gerhard Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), no. 3, 547–569. [F] William J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11. [Ha1] Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. [Ha2] , Convex hypersurfaces with pinched second fundamental form (1982). [Hu1] Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. [Hu2] , Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463–480. [Hu3] , Deforming hypersurfaces of the sphere by their mean curvature, Math. Z. 195 (1987), 205–219. [HP] Gerhard Huisken and Alexander Polder, Geometric evolution equations for hypersurfaces, Calc. of Var. and Geom. Evo. Probl., CIME Lectures of Centraro, springer, (1996). [K] N. E. Krylov, Nonlinear elliptic and parabolic equations of the second order, Reidel, Dordrecht, 1987. [L] Qi-Rui Li, Surfaces expanding by the power of the Gauss curvature flow, Proc. Amer. Math. Soc. 138 (2010), no. 11, 4089–4102. [O] M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213. [S1] Oliver C. Schn¨urer, Surfaces contracting with speed |A|2, J. Differential Geom. 71 (2005), no. 3, 347–363. [S2] , Surfaces expanding by the inverse Gauß curvature flow, J. Reine Angew. Math. 600 (2006), 117–134. [S3] Felix Schulze, Convexity estimates for flows by powers of the mean curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 2, 261–277. 18 BEN ANDREWS AND XUZHONG CHEN
[T] Kaising Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), no. 6, 867–882. [WTL] Chuanxi Wu, Daping Tian, and Guanghan Li, Forced flow by powers of the mth mean curvature, Armen. J. Math. 3 (2010), no. 2, 61–91.
Mathematical Sciences Center, Tsinghua University; and Mathematical Sciences Institute, Aus- tralia National University. E-mail address: [email protected]
Department of Mathematics, East China Normal University E-mail address: [email protected]