Contraction of surfaces in hyperbolic space and in sphere
aa r X i v : . [ m a t h . DG ] A p r CONTRACTION OF SURFACES IN HYPERBOLIC SPACE AND INSPHERE
YINGXIANG HU, HAIZHONG LI, YONG WEI, AND TAILONG ZHOU
Abstract.
In this paper, we consider the contracting curvature flow of smooth closed surfacesin 3-dimensional hyperbolic space and in 3-dimensional sphere. In the hyperbolic case, we showthat if the initial surface M has positive scalar curvature, then along the flow by a positivepower α of the mean curvature H , the evolving surface M t has positive scalar curvature for t >
0. By assuming α ∈ [1 , M t contracts a point in finite timeand become spherical as the final time is approached. We also show the same conclusion for theflows by powers of scalar curvature and by powers of Gauss curvature provided that the power α ∈ [1 / , α of mean curvature contractsstrictly convex surface in S to a round point in finite time if α ∈ [1 , α ∈ [1 / , Introduction
Let R ( c ) ( c = 0 , , −
1) be a real simply connected space form, i.e., when c = 0, R (0) = R ,when c = 1, R (1) = S and when c = − R ( −
1) = H . Let M be a smooth closed surface in R ( c ), which is given by a smooth immersion X : M → R ( c ). We consider the evolution ofclosed surfaces starting at M in R ( c ), according to the following equation of the immersions X : M × [0 , T ) → R ( c ): ∂∂t X ( x, t ) = − F ( x, t ) ν ( x, t ) ,X ( x,
0) = X ( x ) , (1.1)where F is a smooth, symmetric function of the principal curvatures κ = ( κ , κ ) of the evolvingsurface M t = X ( M, t ), and ν is the outward unit normal of M t .1.1. Background.
There are many papers which consider the evolution of hypersurfaces inEuclidean space R n +1 under the flow (1.1). In his foundational work [23], Huisken proved thatany compact strictly convex hypersurface in Euclidean space, evolving by the mean curvatureflow (i.e., F is given by the mean curvature H ), will become spherical as it shrinks to a point.Later, Chow [16] proved the same conclusion for flow (1.1) with speed F given by n -th rootof the Gauss curvature K . He also proved a result for flow by the square root of the scalarcurvature [17], but in that case a stronger assumption than convexity was required for the initialhypersurface. These results have been generalized by Andrews [2, 7, 8] to a large class of speedfunctions F which are homogeneous of degree one of the principal curvatures, and satisfy certainnatural concavity conditions. Date : April 5, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Contracting curvature flow, hyperbolic space, positive scalar curvature, porous mediumequation.
For speed function F with higher homogeneity, the analysis of the flow (1.1) becomes muchmore difficult. In this direction, several works have treated such flows in a special case of surfacesin 3-dimensional Euclidean space R : Andrews [4] proved that any strictly convex surface in R will shrink to a round point along the Gauss curvature flow (i.e., the flow (1.1) with F = K ),which affirmatively resolves the famous Firey’s conjecture. The key step in the proof is acurvature pinching estimate, which says that the ratio of the largest principal curvture to thesmallest one can be controlled by its initial value. This was proved by applying the maximumprinciple to a suitably chosen function of the principal curvatures κ , κ of M t . This idea hasbeen explored further in the works [9, 27, 28]. In particular, Schulze and Schn¨urer[28] provedthat the flow by powers of mean curvature in R (i.e., F = H α ) constracts convex surface to around point provided that α ∈ [1 , R (i.e., F = K α ) contracts convex surface to a round point provided that α ∈ [1 / , G ( x, t ) = ( κ − κ ) κ κ F (1.2)and showed that the spatial maximum of G is monotone non-increasing along the flow for certainrange of the power α , which allows the authors to prove the crucial curvature pinching estimate.In the higher dimensional case, the flow by powers of Gauss curvature has been well studied.The complete picture for this flow has been captured by the combined works [11, 15, 22]. Forgeneral curvature flows with high powers homogeneous speed functions in Euclidean space, inorder to show that closed convex hypersurface shrinks to a round point, a strong curvaturepinching condition on the initial hypersurface is always needed, see [1, 13, 28].In the hyperbolic space, the understanding of the flow (1.1) is less complete. Huisken [24]proved that the mean curvature flow contracts compact hypersurface with principal curvaturessatisfying κ i H > n ( ∀ i = 1 , · · · , n ) in hyperbolic space H n +1 to a round point. Andrews [3]considered a large class of fully nonlinear flows which doesn’t include the mean curvature flow.It is shown that any initial compact hypersurface in hyperbolic space which is horosphericallyconvex (i.e., κ i > i = 1 , · · · , n ) can be deformed to a round point along the flow. Atypical example included in [3] is the flow by shifted harmonic mean curvature F = ( P ni =1 ( κ i − − ) − . In [30], Yu studied the contracting flows in hyperbolic space for a general class ofhomogeneous of degree one speed functions, using a similar argument as in Gerhardt [21] forcurvature flows in the sphere. Recently, Andrews and Chen [10] proved the smooth convergenceof the mean curvature flow for hypersurfaces with positive Ricci curvature in hyperbolic space.In the special case of surfaces in 3-dimensional hyperbolic space H , they also studied thebehavior of the flow (1.1) for surfaces with positive intrinsic scalar curvature R = 2( K − > F = K − R/
2) preserve the condition
R > H produces terms in the evolution of the second fundamental form, which prevent the estimates inEuclidean setting from being applied in hyperbolic setting. To overcome this difficulty, differentimproving quantities have been used in [10] to obtain the curvature pinching estimate.For the curvature flows in the sphere, Huisken [25] proved that for any initial hypersurfacewhich satisfies a pointwise pinching condition, the mean curvature flow will contract the hyper-surface to a point in finite time, or evolve for all time to a smooth totally geodesic hypersurface.Gerhardt [21] proved that any strictly convex hypersurface will be contracted to a round pointin finite time along the flow in sphere if the speed function F is concave and inverse concavewith respect to the principal curvatures. In the special case of surfaces in 3-dimensional sphere ONTRACTION OF SURFACES 3 S , Andrews [5] optimised the choice of the fully nonlinear speed function to show that anysurfaces with positive intrinsic curvature in sphere S can be deformed to either a round pointin finite time, or to the great sphere in infinite time. McCoy [26] recently proved that the flow byany homogeneous of degree one speed function or by Gauss curvature can evolve strictly convexsurfaces to a round point in finite time.1.2. Main result.
In this paper, we focus on the contracting curvature flow of surfaces in3-dimensional hyperbolic space H and in 3-dimensional sphere S . In the hyperbolic case,we assume that the initial surface has positive intrinsic scalar curvature. We will study thecontraction of such surfaces along the flow (1.1) in H by powers of mean curvature, powers ofscalar curvature and powers of Gauss curvature. Our first result states as follows: Theorem 1.1.
Let X : M → H be a smooth closed surface with positive scalar curvature in H . Assume that either (i) F = H α with α ∈ [1 , ; or (ii) F = ( K − α with α ∈ [1 / , ; or (iii) F = K α with α ∈ [1 / , .Then there exists a unique solution X : M × [0 , T ) → H of the flow (1.1) on a maximum timeinterval [0 , T ) , where T < ∞ . The surfaces M t has positive scalar curvature for each t ∈ [0 , T ) ,and converges smoothly to a point p ∈ H as t → T . The solutions are asymptotic to a shrinkingsphere as t → T in the following sense: Let Θ( t, T ) be the spherical solution of the flow with thesame existence time T . Introducing geodesic polar coordinate system with respect to the point p , and writing the evolving surfaces M t as graphs of a function u ( x, t ) on S , then the rescaledfunction u Θ − is uniformly bounded and converges to in C ∞ ( S ) as t → T . In the sphere case, we assume that the initial surface is strictly convex in S . Then it liesstrictly in a hemisphere of S . We consider its contraction along the flow (1.1) in S by powersof mean curvature and powers of Gauss curvature. Theorem 1.2.
Let X : M → S be a smooth, closed and strictly convex surface in S . Assumethat either (i) F = H α with α ∈ [1 , ; or (ii) F = K α with α ∈ [1 / , .Then there exists a unique solution X : M × [0 , T ) → S of the flow (1.1) on a maximumtime interval [0 , T ) , where T < ∞ . The surface M t is strictly convex for each t ∈ [0 , T ) , andconverges smoothly to a point p ∈ S as t → T . The solutions are asymptotic to a shrinkingsphere as t → T in the following sense: Let Θ( t, T ) be the spherical solution of the flow with thesame existence time T . Introducing geodesic polar coordinate system with respect to the point p , and writing the evolving surfaces M t as graphs of a function u ( x, t ) on S , then the rescaledfunction u Θ − is uniformly bounded and converges to in C ∞ ( S ) as t → T . Outline of the proof.
As the first step, we need to show that the scalar curvature ofthe evolving surface M t remains to be positive along the flow (1.1) in H . This is equivalent toshow that the Gauss curvature K > M t for t >
0, and can follow from the evolutionequation of K and parabolic maximum principle. For the flow in sphere S , we also apply themaximum principle to the evolution equation of K and obtain that K is greater than its initialvalue. This implies the strict convexity of the surfaces M t is preserved along the flow (1.1) for t > Y. HU, H. LI, Y. WEI, AND T. ZHOU
The key step in the proof our main theorems is to prove the curvature pinching estimate. Inthe hyperbolic case, we consider the following function G ( x, t ) := ( κ − κ ) ( κ κ − F (1.3)on the evolving surfaces M t (inspired by [10]). Since K > G iswell-defined on M t for all t ∈ [0 , T ). We will show that the spatial maximum of G is monotonenon-increasing along the flow (1.1) with speeds listed in Theorem 1.1. The proof is by applyingmaximum principle to the evolution equation of G . The main advantage of this quantity G isthat there are no zero order terms in its evolution equation, provided that the speed function F is a homogeneous function of the principal curvatures or F is a power of the scalar curvature.Thus, the monotonicity of G along the flow reduces to the non-positivity of the gradient termsat the critical points. This would be the most technique part in the proof: We will first derive in § G for general speedfunction F . Then in § §
5, we treat the three speed functions listed in Theorem 1.1 separately.By careful calculation, we eventually find the range for the power α such that the gradient termsare non-positive, and therefore we conclude that along the flow the spatial maximum of G wouldbe monotone non-increasing in time. We should point out that the analysis of the gradient termshere is much more complicated due to the negative curvature of the ambient space H . In fact,the coefficients in front of ( ∇ h ) and ( ∇ h ) are not homogeneous in κ , κ , which is quitedifferent from the Euclidean case considered in [28].To prove the curvature pinching estimate in S , we use the same test function as in Euclideancase [9, 28]. That is, we use the function G defined in (1.2). The evolution equation for G has both the zero order terms and gradient terms. Since the function G is the same one asin Euclidean case used in [9, 28], the analysis of the gradient terms would be similar with theEuclidean case. We will apply the estimate of gradient terms in [28] directly here for the flowby powers of mean curvature in the sphere. However, the estimate of the gradient terms in [9]for the flow by powers of Gauss curvature is carried out using the Gauss map parametrization ofthe flow: The flow of surfaces in Euclidean space R is equivalent to a scalar parabolic equationon the sphere S for the support function of the evolving surfaces. This parametrization is notavailable in the sphere case. Instead we prove our estimate using the calculation on the evolvingsurfaces directly. Moreover, due to the positive curvature of the ambient space S we have agood sign for the zero order terms. Thus the maximum principle can be applied to give themonotonicity of the function G . The details will be given in § H by powers of mean curvature, since theproof is similar for the remaining flows. We first observe that the maximum existence timeis finite, this follows from an estimate on the mean curvature from below. By the curvaturepinching estimate obtained in §
3, an argument as in [21, §
6] implies that the ratio of the outerradius of the domain enclosed by M t to the inner radius is bounded for t sufficiently close to T . We choose a time t such that the above estimate holds for all t ∈ [ t , T ). An argumentof Tso [29] would imply that the mean curvature remains bounded as long as M t encloses anon-vanishing volume. On the other hand, an upper bound on mean curvature together withthe lower bound H ≥ √ K > t → T and the flow remains smoothuntil it contracts to a point p as t → T . ONTRACTION OF SURFACES 5
To describe the asymptotical behavior of the flow as the final time is approached, we studythe convergence of appropriately rescaled flows using a similar program described in [21] forcontracting curvature flow in the sphere. For α = 1, the H α -flow (3.1) is just the mean curvatureflow studied by Andrews and Chen [10]. So we focus on α ∈ (1 , t, T ) be the sphericalsolution of (3.1) in the hyperbolic space H which shrinks to a point as t → T . Introducea geodesic polar coordinate around the point p the flow are shrinking to, and write M t =graph u ( x, t ) on the sphere S for t ∈ [ t , T ). We consider the rescaling ˜ H = H Θ( t, T ) on themean curvature, and ˜ u = u Θ − ( t, T ) on the function u . The upper bound on ˜ H can be provedusing the technique of Tso. However, for α > H has a coefficient α ˜ H α − in the second order part and it becomes degenerate for ˜ H sufficiently small. Thus we can not apply the parabolic Harnack inequality as in [21] to get anestimate from below for ˜ H . To overcome this problem, we write the evolution equation satisfiedby ˜ H as a porous medium equation, and apply a result of DiBenedetto and Friedman [18] toget the H¨older continuity of ˜ H . Moreover, for α > § ≤ κ κ ≤ C ˜ H (1 − α ) Θ ( α − ( t, T )This would be enough for our purpose to show the convergence of the rescaled function ˜ u to 1as t → T . See § Acknowledgments.
The authors would like to thank Professor Ben Andrews for suggestionson the curvature pinching in the hyperbolic case. The first author was supported by ChinaPostdoctoral Science Foundation (No.2018M641317). The second author was supported byNSFC grant No.11671224, 11831005 and NSFC-FWO grant No.1181101212. The third authorwas supported by Discovery Early Career Researcher Award DE190100147 of the AustralianResearch Council. 2.
Evolution equations
In this section, we collect some basic evolution equations along the flow (1.1), and then derivethe evolution equation of the quantity G defined in (1.3).2.1. Notations.
First, we fix the notations we will use in the paper. We denote by g = ( g ij ) and h = ( h ij ) the induced metric and the second fundamental form of the surface M t , respectively.Then the Weingarten tensor W = ( h ji ) = ( g jk h ik ). The eigenvalues κ , κ of W are calledthe principal curvatures of M t . In the flow (1.1), the speed function F = F ( W ) is a smoothsymmetric function of the Weingarten tensor W = ( h ji ) of the evolving surface M t = X t ( M ),and ν is the outward unit normal of M t . Equivalently, F = F ( W ) = f ( κ ( W )), where f is asmooth symmetric function of 2-variables, and κ ( W ) = ( κ , κ ) denotes the eigenvalues of W .The derivatives of F with respect to the components of W = ( h ji ) and those of f with respectto κ i are related in the following way (see e.g. [7]): Write˙ f i = ∂f∂κ i , ¨ f ij = ∂ f∂κ i ∂κ j . to denote the derivatives of f with respect to κ i . If A is diagonal and B is a symmetric matrix,then the first derivative of F is given by˙ F kl ( A ) = ˙ f k ( κ ( A )) δ kl ; (2.1) Y. HU, H. LI, Y. WEI, AND T. ZHOU and if A has distinct eigenvalues, then the second derivatives of F in direction B is given by¨ F kl,rs B kl B rs = ¨ f kl ( κ ( A )) B kk B ll + 2 X k Along the flow (1.1) in R ( c ), we have the following evolutionequations (see [3, 20]): ∂∂t g ij = − F h ij , (2.3) ∂∂t F = ˙ F ij ∇ i ∇ j F + F ˙ F ij (( h ) ij + cg ij ) , (2.4)where ( h ) ij = h ri h rj . The Weingarten matrix W = ( h ji ) satisfies the following parabolic equa-tion: ∂∂t h ji = ˙ F kl ∇ k ∇ l h ji + ¨ F kl,pq ∇ i h kl ∇ j h pq + ˙ F kl ( h rk h rl − cg kl ) h ji + ( F − ˙ F kl h kl ) h pi h jp + c ( F + ˙ F kl h kl ) δ ji . (2.5)For any smooth symmetric function G = G ( h ji ) of the Weingarten tensor, we have (see [3]) ∂∂t G = ˙ F ij ∇ i ∇ j G + (cid:16) ˙ G ij ¨ F kl,mn − ˙ F ij ¨ G kl,mn (cid:17) ∇ i h kl ∇ j h mn + ( F − ˙ F kl h kl ) ˙ G ij ( h ) ij + ˙ F kl ( h ) kl ˙ G ij h ij + c (cid:16) ( F + ˙ F kl h kl ) ˙ G ij g ij − ˙ F kl g kl ˙ G ij h ij (cid:17) . (2.6)The key step in the proof of our results is to obtain the curvature pinching estimate of theflow, i.e., we prove that the ratio of the largest principal curvature κ to the smallest one κ is controlled by its initial value. In hyperbolic space case, we will prove this estimate byapplying maximum principle to the evolution equation of the function G defined in (1.3), whichis obviously a smooth symmetric function G = G ( W ) of the Weingarten tensor W : G = G ( W ) = | h | − K ( K − F As before we can equivalently write G = g ( κ ), where g is a smooth symmetric function of theprincipal curvatures. The following proposition shows that the function G (defined in (1.3))has the advantage that its evolution equation (2.6) along the flow (1.1) in hyperbolic space hasno zero-order terms, in the cases that the speed function F is homogeneous of the principalcurvatures, or the powers of the scalar curvature. Proposition 2.1. Let α > . Let M t be a smooth solution to the flow (1.1) with positivescalar curvature in H . Assume either (i). the speed function F is a homogeneous function ofthe principal curvatures, or (ii). F = ( K − α is a power of the scalar curvature. Then theevolution of G defined in (1.3) satisfies ∂∂t G = ˙ F ij ∇ i ∇ j G + (cid:16) ˙ G ij ¨ F kl,mn − ˙ F ij ¨ G kl,mn (cid:17) ∇ i h kl ∇ j h mn . (2.7) ONTRACTION OF SURFACES 7 Proof. Since the function G in (1.3) is a smooth symmetric function of W , we have the evolutionequation (2.6) for G along the flow (1.1). Write F = f ( κ ) , G = g ( κ ) = ( κ − κ ) ( κ κ − f ( κ ) . We have ˙ g = 2( κ − κ ) fκ κ − ( κ − κ ) ˙ f κ κ − κ − f ( κ κ − ! , (2.8)˙ g = 2( κ − κ ) fκ κ − ( κ − κ ) ˙ f κ κ − − κ ) f ( κ κ − ! . (2.9)A direct calculation gives ˙ g + ˙ g =2 gf ( ˙ f + ˙ f ) − g ( κ + κ ) κ κ − , (2.10)˙ g κ + ˙ g κ =2( α + 1) g − gκ κ κ κ − , (2.11)˙ g κ + ˙ g κ =2 gf ( ˙ f κ + ˙ f κ ) − g ( κ + κ ) κ κ − . (2.12)(i). If F is a homogeneous of degree α function of the principal curvatures, we have the Eulerrelation ˙ f κ + ˙ f κ = αf . Using (2.1), the zero-order term of (2.6) for G can be computed asfollows: Q =( F − X i =1 ˙ f i κ i ) X i =1 ˙ g i κ i + ( X i =1 ˙ f i κ i + X i =1 ˙ f i ) X i =1 ˙ g i κ i − ( F + X i =1 ˙ f i κ i ) X i =1 ˙ g i =(1 − α )( ˙ g κ + ˙ g κ ) f + ( ˙ g κ + ˙ g κ )( ˙ f κ + ˙ f κ ) − ( α + 1)( ˙ g + ˙ g ) f + ( ˙ g κ + ˙ g κ )( ˙ f + ˙ f ) . (2.13)Substituting the equations (2.10) – (2.12) into (2.13), we get Q =(1 − α ) (cid:20) g ( ˙ f κ + ˙ f κ ) − f g ( κ + κ ) κ κ − (cid:21) + ( ˙ f κ + ˙ f κ ) (cid:20) α + 1) g − gκ κ κ κ − (cid:21) − ( α + 1) (cid:20) g ( ˙ f + ˙ f ) − f g ( κ + κ ) κ κ − (cid:21) + ( ˙ f + ˙ f ) (cid:20) α + 1) g − gκ κ κ κ − (cid:21) = − gκ κ − h ( ˙ f κ + ˙ f κ ) − αf ( κ + κ ) + κ κ ( ˙ f + ˙ f ) i = − gκ κ − h ( κ + κ )( ˙ f κ + ˙ f κ ) − αf ( κ + κ ) i =0 . Y. HU, H. LI, Y. WEI, AND T. ZHOU (ii). If F = ( K − α , the first order derivatives of F and G are given by:˙ f = α ( K − α − κ , ˙ f = α ( K − α − κ ˙ g =2( κ − κ )( K − α − + 2( α − κ − κ ) ( K − α − κ ˙ g = − κ − κ )( K − α − + 2( α − κ − κ ) ( K − α − κ . Substituting these equations into the zero order term Q of the equation (2.6), we also have Q = 0. The equation (2.7) follows immediately. (cid:3) In the sphere case, we will prove the curvature pinching estimate by choosing the test function G as defined in (1.2). The evolution equation for G has both the zero order terms and gradientterms. Since the function G in (1.2) is the same one as in Euclidean case used in [9, 28], theanalysis of the gradient terms would be similar as the Euclidean case. We will describe this indetails in § Computation of the gradient terms. Now we calculate the gradient term of (2.7)explicitly. Suppose p is a point in M where a new spatial maximum of G is attained at time t ∈ [0 , T ). Choose local orthonormal coordinates for M near p such that h ij ( p, t ) = diag( κ , κ ).By (2.1) and (2.2), the gradient term on the RHS of the evolution equation (2.7) for G can beexpressed as follows: Q = (cid:16) ˙ G ij ¨ F kl,mn − ˙ F ij ¨ G kl,mn (cid:17) ∇ i h kl ∇ j h mn = (cid:16) ˙ g ¨ f − ˙ f ¨ g (cid:17) ( ∇ h ) + (cid:16) ˙ g ¨ f − ˙ f ¨ g (cid:17) ( ∇ h ) + 2 (cid:16) ˙ g ¨ f − ˙ f ¨ g (cid:17) ∇ h ∇ h + (cid:16) ˙ g ¨ f − ˙ f ¨ g (cid:17) ( ∇ h ) + (cid:16) ˙ g ¨ f − ˙ f ¨ g (cid:17) ( ∇ h ) + 2 (cid:16) ˙ g ¨ f − ˙ f ¨ g (cid:17) ∇ h ∇ h + 2 ˙ g ˙ f − ˙ g ˙ f κ − κ ( ∇ h ) + 2 ˙ g ˙ f − ˙ g ˙ f κ − κ ( ∇ h ) . (2.14)Without loss of generality, we can assume that G is nonzero at ( p, t ) (otherwise M t is a sphereand the proof is trivial). We can also assume that κ > κ because the maximum point of G isnot umbilical and both F and G are smooth and symmetric.At the spatial maximum point of G , the gradient conditions ∇ i G = 0 give two equations:˙ g ∇ h + ˙ g ∇ h = 0 , ˙ g ∇ h + ˙ g ∇ h = 0 . (2.15)For simplicity, we denote β =( κ − κ )( κ κ − 1) ˙ f + ( κ − f, (2.16) γ =( κ − κ )( κ κ − 1) ˙ f − ( κ − f. (2.17)Then ˙ g = 2( κ − κ ) f ( κ κ − β, ˙ g = 2( κ − κ ) f ( κ κ − γ. (2.18)Assume that at least one of ˙ g and ˙ g does not vanish. As M t is a family of surfaces with positivescalar curvature, i.e., κ κ > 1, we have(i) If ˙ g = 0, then β = 0 and γβ = ˙ g ˙ g ; ONTRACTION OF SURFACES 9 (ii) If ˙ g = 0, then γ = 0 and βγ = ˙ g ˙ g .Now we define T := ( ∇ h ) γ , if γ = 0;( ∇ h ) β , if β = 0; T := ( ∇ h ) γ , if γ = 0;( ∇ h ) β , if β = 0; (2.19)Substituting (2.15) into (2.14) and using Codazzi equation, with the notation (2.19) we obtain Q = (cid:18) ( ˙ g ¨ f − ˙ f ¨ g ) γ − 2( ˙ g ¨ f − ˙ f ¨ g ) βγ + (cid:0) ˙ g ¨ f − ˙ f ¨ g + 2 ˙ g ˙ f − ˙ g ˙ f κ − κ (cid:1) β (cid:19) T + (cid:18) ( ˙ g ¨ f − ˙ f ¨ g ) β − 2( ˙ g ¨ f − ˙ f ¨ g ) βγ + (cid:0) ˙ g ¨ f − ˙ f ¨ g + 2 ˙ g ˙ f − ˙ g ˙ f κ − κ (cid:1) γ (cid:19) T . (2.20)We only calculate the coefficient (denoted by Z ) in front of T , since the coefficient in front of T is similar, just with κ and κ interchanged. Z :=( ˙ g ¨ f − ˙ f ¨ g ) γ − 2( ˙ g ¨ f − ˙ f ¨ g ) βγ + ˙ g ¨ f − ˙ f ¨ g + 2 ˙ g ˙ f − ˙ g ˙ f κ − κ ! β . (2.21)Taking the further derivatives to (2.8) and (2.9), we get the second derivatives of g :¨ g = 2( κ − f ( κ κ − − κ ( κ − κ )( κ − f ( κ κ − + 8( κ − κ )( κ − f ˙ f ( κ κ − + 2( κ − κ ) ( ˙ f ) ( κ κ − + 2( κ − κ ) f ¨ f ( κ κ − , (2.22)¨ g = 2( κ − f ( κ κ − − κ ( κ − κ )( κ − f ( κ κ − + 8( κ − κ )( κ − f ˙ f ( κ κ − + 2( κ − κ ) ( ˙ f ) ( κ κ − + 2( κ − κ ) f ¨ f ( κ κ − , (2.23)¨ g = − f ( κ κ − + 6( κ − κ ) f ( κ κ − + 2( κ − κ ) ˙ f ˙ f ( κ κ − + 4( κ − κ ) f [ ˙ f ( κ − − ˙ f ( κ − κ κ − + 2( κ − κ ) f ¨ f ( κ κ − . (2.24)By (2.8)-(2.9) and (2.22)-(2.24), we get˙ g ˙ f − ˙ g ˙ f κ − κ = − f h ( κ − 1) ˙ f + ( κ − 1) ˙ f i ( κ κ − , (2.25) and ˙ g ¨ f − ˙ f ¨ g = − κ − κ )( κ − f ( ˙ f ) ( κ κ − − κ − κ ) ( ˙ f ) ( κ κ − − κ − f ˙ f ( κ κ − + 6 κ ( κ − κ )( κ − f ˙ f ( κ κ − + 2( κ − κ )( κ − f ¨ f ( κ κ − , (2.26)˙ g ¨ f − ˙ f ¨ g = − κ − κ )( κ − f ˙ f ˙ f ( κ κ − − κ − κ ) ˙ f ( ˙ f ) ( κ κ − − κ − f ˙ f ( κ κ − + 6 κ ( κ − κ )( κ − f ˙ f ( κ κ − + 2( κ − κ )( κ − f ¨ f ( κ κ − , (2.27)˙ g ¨ f − ˙ f ¨ g = 4( κ − κ ) f ˙ f ( κ κ − (cid:16) ˙ f ( κ − − ˙ f ( κ − (cid:17) + 2 f ˙ f ( κ κ − − κ − κ ) f ˙ f ( κ κ − + 2( κ − κ − κ ) f ¨ f ( κ κ − . (2.28)Then it follows from (2.25) – (2.28) that Z = 2( κ − κ )( κ − f ( κ κ − (cid:16) ¨ f γ + ¨ f β − f γβ (cid:17) + 4 f (cid:16) ( κ − 1) ˙ f + ( κ − 1) ˙ f (cid:17) κ κ − (cid:18) − κ − κ ) f ˙ f − ( κ − f ( κ κ − + ( κ − κ ) ˙ f ˙ f (cid:19) + 4( κ − κ ) f ˙ f κ κ − (cid:18) − κ − κ )( κ − f ˙ f κ κ − 1+ ( κ − κ − f ( κ κ − − ( κ − κ ) ˙ f ˙ f (cid:19) . (2.29)3. Flow in H by powers of mean curvature In this section, we study the flow by powers of mean curvature in the hyperbolic space H .Let X : M → H be a smooth closed surface in H . We consider a family of closed surfaces M t = X ( M, t ) in H satisfying ∂∂t X ( x, t ) = − H α ( x, t ) ν ( x, t ) , α > X ( x, 0) = X ( x ) . (3.1)We first prove that K > H α -flow (3.1) in H . Proposition 3.1. Let M t , t ∈ [0 , T ) be the smooth solution to the flow (3.1) in H . For anypower α > , if the Gauss curvature K > at t = 0 , then K > on M t for all t ∈ [0 , T ) . ONTRACTION OF SURFACES 11 Moreover, we have K − ≥ min M ( K − (cid:18) − α ( α + 1) t min M ( K − α +12 (cid:19) − α +1 (3.2) on M t for t ∈ [0 , T ) .Proof. Let F = H α = f ( κ ) = ( κ + κ ) α . Then˙ f = ˙ f = αH α − , ¨ f = ¨ f = ¨ f = α ( α − H α − . We apply (2.6) to calculate the evolution equation of the Gauss curvature K along the flow(3.1). Let G = K in (2.6). We have ∂∂t G = ˙ F ij ∇ i ∇ j G + Q + Q , (3.3)where we denote Q , Q the gradient term and zero order term in the evolution of G . Since G = g ( κ ) = κ κ , the derivatives of G are given by˙ g = κ , ˙ g = κ , ¨ g = ¨ g = 0 , ¨ g = ¨ g = 1 . Then ˙ G ij ( h ) ij = KH, ˙ G ij h ij = 2 K, ˙ G ij g ij = H. A direct calculation gives that the zero order term Q satisfies Q =(1 − α ) KH α +1 + 2 K ( | A | + 2) αH α − − ( α + 1) H α +1 =( K − (cid:0) H + α ( κ − κ ) (cid:1) H α − . (3.4)At the spatial minimum point of K , we have ∇ K = ∇ K = 0, which implies that κ ∇ i h + κ ∇ i h = 0 , i = 1 , . By the general formula (2.14), we obtained that the gradient term Q at the spatial minimumpoint of K : Q = α ( α − H α − κ (cid:0) ( ∇ h ) + 2 ∇ h ∇ h + ( ∇ h ) (cid:1) + α ( α − H α − κ (cid:0) ( ∇ h ) + 2 ∇ h ∇ h + ( ∇ h ) ) (cid:1) − αH α − (cid:0) ∇ h ∇ h − ( ∇ h ) + ∇ h ∇ h − ( ∇ h ) (cid:1) = ακ κ − H α − (cid:0) ( α − κ − κ ) + 2 H (cid:1) ( ∇ h ) + ακ κ − H α − (cid:0) ( α − κ − κ ) + 2 H (cid:1) ( ∇ h ) (3.5)Applying maximum principle to (3.3) and using (3.4) and (3.5), we have ddt min M t ( K − ≥ ( K − (cid:0) H + α ( κ − κ ) (cid:1) H α − , (3.6)which implies that K > H = κ + κ ≥ √ K and α > 0, the inequality (3.6) implies ddt min M t ( K − ≥ α +1 ( K − α +12 . (3.7)The estimate (3.2) follows by integrating (3.7) in time. (cid:3) Note that the estimate (3.2) implies an upper bound for the maximum existence time T : T ≤ − α α + 1 min M ( K − − α +12 . (3.8)To show the pinching estimate of the principal curvatures of M t , we consider the followingquantity on M t : G ( x, t ) = ( κ + κ ) α ( κ − κ ) ( κ κ − . (3.9)Since K = κ κ > M t for all t ∈ [0 , T ), the quantity G is well-defined. Firstly, we deducethe evolution equation of G along the H α -flow (3.1). Lemma 3.2. The evolution equation of G along the H α -flow (3.1) satisfies ∂∂t G = αH α − ∆ G + 2 αH α − ( κ κ − ( a T + a T ) (3.10) at the spatial maximum point, where ∆ denotes the Laplacian operator with respect to the metric g ( t ) on M t , T and T are defined as (2.19) , and the coefficients a , a are given by a ( α, κ , κ ) := 4( κ − κ ) ( κ κ − · α − ( κ − κ )( κ + κ ) (cid:18) (3 κ + 1) κ + 4 κ ( κ − κ + 2( − κ + κ + 2) κ + 4 κ ( κ − κ + 2) κ + ( − κ + κ − (cid:19) · α (3.11)+ ( κ + κ )( κ − (cid:18) κ − κ κ + (4 − κ ) κ + 2 κ κ + ( − κ + 12 κ − κ − κ (cid:19) and a ( α, κ , κ ) := a ( α, κ , κ ) .Proof. Firstly, since F = H α , the second order term in (2.7) becomes ˙ F ij ∇ i ∇ j G = αH α − ∆ G .This is just the first term of (3.10). We next compute the gradient terms. Note that ˙ g , ˙ g cannot vanish at the same time. Using the equations (2.16) - (2.18) and substituting F = H α , wehave ( ˙ g ) + ( ˙ g ) = 2( κ − κ ) H α ( κ κ − ( β + γ ) , where β := αH α − ( κ κ − κ − κ ) + H α ( κ − ,γ := αH α − ( κ κ − κ − κ ) − H α ( κ − . If β = γ = 0, then κ + κ − κ + κ − ≥ κ κ − > . Therefore ˙ g , ˙ g can not vanish at the same time.By the general formula (2.29), the coefficient in front of T is equal to Z = 2 α ( α − H α − ( κ − κ )( κ − κ κ − ( γ − β ) + 4 αH α − ( κ + κ − κ κ − (cid:18) − κ − κ ) αH α − − ( κ − H α ( κ κ − + ( κ − κ ) α H α − (cid:19) + 4 αH α − ( κ − κ ) κ κ − (cid:18) − κ − κ )( κ − αH α − κ κ − 1+ ( κ − κ − H α ( κ κ − − ( κ − κ ) α H α − (cid:19) = 2 α ( α − H α − ( κ − κ )( κ − κ + κ − ( κ κ − + 4 αH α − ( κ + κ − κ κ − (cid:18) − κ − κ ) αH − ( κ − H ( κ κ − + ( κ − κ ) α (cid:19) + 4 αH α − ( κ − κ ) κ κ − (cid:18) − κ − κ )( κ − αHκ κ − 1+ ( κ − κ − H ( κ κ − − ( κ − κ ) α (cid:19) . Finally, we obtain a := ( κ κ − αH α − Z = ( α − κ − κ )( κ − κ + κ − H − αH ( κ − κ )( κ κ − ( κ + κ − − αH ( κ − κ ) ( κ − κ κ − − κ − ( κ + κ − H + 2( κ − κ ) ( κ − κ − H + 4 α ( κ − κ ) ( κ κ − . Rewrite it as a quadratic polynomial of α , we obtain the desired expression (3.11). (cid:3) Now we apply the maximum principle to the evolution equation (3.10) to prove the mono-tonicity of G for suitable range of α . Theorem 3.3. Let M t be a smooth solution of (3.1) in H with K > , where α ∈ [1 / , , then max M t G ( · , t ) is non-increasing in time.Proof. To this theorem, we need to show the coefficients a , a of the gradient terms are non-positive at the spatial maximum point of G . At the spatial maximum point ( p, t ) of G , weassume that G is nonzero (otherwise M t is a sphere and the proof is trivial). Without loss ofgenerality, we further assume that κ > κ . Define a convex subset C of R by C := { ( κ , κ ) ∈ R | κ κ > , κ > κ } . By (3.11), the coefficients a i ( α, κ , κ ), i = 1 , α for any fixedpoint ( κ , κ ) ∈ C . We claim that for all ( κ , κ ) ∈ C , a i (1 / , κ , κ ) ≤ , a i (4 , κ , κ ) ≤ . This would imply that a i ( α, κ , κ ) ≤ κ , κ ) ∈ C provided that α ∈ [1 / , h i,α ( x ) := a i ( α, x, κ ), where ( x, κ ) ∈ C . It suffices to showthat h i, / ( κ ) ≤ , h i, ( κ ) ≤ . Note that for any ( κ , κ ) ∈ C , we have κ > max { κ , /κ } . Claim 1: h i, / ( κ ) ≤ i = 1 , h ( k ) i,α ( x ) the k -th derivative of h i,α ( x ) with respect to x . Clearly, we have h (6)1 , / ( x ) = − < , h (7)2 , / ( x ) = − κ + 4 x ) < , for all x > κ > κ ≥ κ , then κ ≥ 1; The derivatives of h i, / ( x ) at x = κ satisfy h (5)1 , / ( κ ) = − κ (9 + 2 κ ) ≤ h (4)1 , / ( κ ) = − − κ + 7 κ ) ≤ h (3)1 , / ( κ ) = − κ ( κ − κ − ≤ h (2)1 , / ( κ ) = − 329 ( κ − (29 κ − ≤ h (1)1 , / ( κ ) = − κ ( κ − (8 κ − ≤ h , / ( κ ) = − κ ( κ − ≤ . and h (6)2 , / ( κ ) = − − κ ) ≤ h (5)2 , / ( κ ) = − κ ( − 51 + 338 κ ) ≤ h (4)2 , / ( κ ) = − κ ( − 79 + 187 κ ) ≤ h (3)2 , / ( κ ) = − κ ( − κ )( − κ ) ≤ h (2)2 , / ( κ ) = − 329 ( κ − − − κ + 113 κ ) ≤ h (1)2 , / ( κ ) = − κ ( κ − (16 κ − ≤ h , / ( κ ) = − κ ( κ − ≤ . (ii) If κ ≤ κ , then 0 < κ ≤ 1; The derivatives of h i, / ( x ) at x = 1 /κ satisfy h (5)1 , / (1 /κ ) = − κ (18 − κ + 2 κ ) ≤ h (4)1 , / (1 /κ ) = − κ (45 − κ − κ + 11 κ ) ≤ ONTRACTION OF SURFACES 15 h (3)1 , / (1 /κ ) = − κ (1 − κ )(5 + 4 κ )(6 − κ − κ ) ≤ h (2)1 , / (1 /κ ) = − κ (1 − κ ) (15 + 24 κ + 3 κ + 2 κ ) ≤ h (1)1 , / (1 /κ ) = − κ (1 − κ ) (6 + 13 κ − κ − κ + 2 κ ) ≤ h , / (1 /κ ) = − κ (1 − κ ) (1 + κ )(1 + 4 κ + κ ) ≤ . and h (6)2 , / (1 /κ ) = − κ (28 + 13 κ + 2 κ ) ≤ h (5)2 , / (1 /κ ) = − κ (168 + 108 κ + 3 κ + 8 κ ) ≤ h (4)2 , / (1 /κ ) = − κ (210 + 165 κ − κ + 13 κ + 11 κ ) ≤ h (3)2 , / (1 /κ ) = − κ (84 + 75 κ − κ + 4 κ + 13 κ + κ ) ≤ h (2)2 , / (1 /κ ) = − κ (1 − κ )(28 + 55 κ − κ − κ − κ ) ≤ h (1)2 , / (1 /κ ) = − κ (1 − κ ) (8 + 24 κ − κ − κ − κ ) ≤ h , / (1 /κ ) = − κ (1 − κ ) (1 + κ )(1 + 4 κ + κ ) ≤ . In both cases, we have h i, / ( κ ) ≤ Claim 2: h i, ( κ ) ≤ i = 1 , h (6)1 , ( x ) = − κ ) < , h (7)2 , ( x ) = − x − κ ) < , for all x > κ > κ ≥ κ , then κ ≥ 1; The derivatives of h i, at x = κ : h (5)1 , ( κ ) = − κ ( κ − ≤ h (4)1 , ( κ ) = − κ + 6 κ ) ≤ h (3)1 , ( κ ) = − κ ( κ − κ + 1) ≤ h (2)1 , ( κ ) = − κ − (3 κ + 1) ≤ h (1)1 , ( κ ) = − κ ( κ − (9 κ − ≤ h , ( κ ) = − κ ( κ − ≤ . and h (6)2 , ( κ ) = − − κ ) ≤ h (5)2 , ( κ ) = − κ ( − 15 + 47 κ ) ≤ h (4)2 , ( κ ) = − κ (25 + 11 κ ) ≤ h (3)2 , ( κ ) = − κ ( − 51 + 60 κ + 7 κ ) ≤ h (2)2 , ( κ ) = − κ − − κ + κ ) ≤ h (1)2 , ( κ ) = − κ ( κ − (3 + 7 κ ) ≤ h , ( κ ) = − κ ( κ − ≤ . (ii) If κ ≤ κ , then 0 < κ ≤ 1; The derivatives of h i, at x = 1 /κ : h (5)1 , (1 /κ ) = − κ (1 − κ )(5 + 8 κ ) ≤ h (4)1 , (1 /κ ) = − κ (75 − κ − κ + 99 κ ) ≤ h (3)1 , (1 /κ ) = − κ (1 − κ )(25 − κ + 6 κ + 15 κ ) ≤ h (2)1 , (1 /κ ) = − κ (1 − κ ) (75 − κ + 37 κ + 21 κ ) ≤ h (1)1 , (1 /κ ) = − κ (1 − κ ) (15 − κ + 7 κ + 9 κ − κ ) ≤ h , (1 /κ ) = − κ (1 − κ ) (1 + κ )(5 − κ + 5 κ ) ≤ . and h (6)2 , (1 /κ ) = − κ (140 − κ + 21 κ ) ≤ h (5)2 , (1 /κ ) = − κ (140 − κ + 63 κ − κ ) ≤ h (4)2 , (1 /κ ) = − κ (350 − κ + 315 κ − κ + 99 κ ) ≤ h (3)2 , (1 /κ ) = − κ (70 − κ + 105 κ − κ + 9 κ − κ ) ≤ h (2)2 , (1 /κ ) = − κ (1 − κ )(140 − κ + 128 κ + 46 κ − κ − κ ) ≤ h (1)2 , (1 /κ ) = − κ (1 − κ ) (20 − κ + 9 κ + 9 κ − κ ) ≤ h , (1 /κ ) = − κ (1 − κ ) (1 + κ )(5 − κ + 5 κ ) ≤ . In both cases, we have h i, ( κ ) ≤ G . Applying the maximum principle we conclude that G is monotone non-increasingalong the H α -flow with α ∈ [1 / , (cid:3) We have proved that max M t G ( x, t ) is monotone non-increasing along the H α -flow for α ∈ [1 / , α ∈ [1 , M t . ONTRACTION OF SURFACES 17 Corollary 3.4. Let M t , t ∈ [0 , T ) be a smooth solution of H α -flow with K > in H , where α ∈ [1 , . There exists a constant C > depending only on the initial surface M and α suchthat < C ≤ κ κ ≤ C, (3.12) on M t for all t ∈ [0 , T ) .Proof. By the monotonicity of G , we have( κ + κ ) α ( κ − κ ) ( κ κ − ≤ C := max M G ( · , . If α ≥ 1, then κ κ + κ κ − κ + κ ) α ( κ − κ ) ( κ κ − ( κ κ − κ κ ( κ + κ ) α ≤ − α C ( κ κ − − α , which is bounded from above by (3.2) and α ≥ 1. The estimate (3.12) follows immediately. (cid:3) Flow in H by powers of scalar curvature In this section, we consider the flow for surfaces in H by powers of scalar curvature, i.e., ∂∂t X ( x, t ) = − ( K ( x, t ) − α ν ( x, t ) , α > X ( x, 0) = X ( x ) . (4.1) Proposition 4.1. Let M t , t ∈ [0 , T ) be the smooth solution of (4.1) in H . For any power α > , if K > on M , then K > on M t for all t ∈ [0 , T ) . Moreover, we have the estimate K − ≥ min M ( K − (cid:18) − (2 α + 1) min M ( K − α + t (cid:19) − α +1 (4.2) on M t for t ∈ [0 , T ) .Proof. By (2.4), the speed function F = ( K − α of the flow (4.1) satisfies the following evolutionequation ∂∂t ( K − α = α ( K − α − ˙ K ij ∇ i ∇ j ( K − α + α ( K − α − ˙ K ij (cid:0) ( h ) ij − g ij (cid:1) = α ( K − α − ˙ K ij ∇ i ∇ j ( K − α + α ( K − α H. (4.3)By applying the maximum principle to (4.3), the spatial minimum of K − t > 0. Then the equation (4.3) is equivalent to ∂∂t ( K − 1) = ˙ K ij ∇ i ∇ j ( K − α + ( K − α +1 H, which implies ddt min M t ( K − ≥ K − α + . (4.4)Integrating (4.4) gives the estimate (4.2). (cid:3) Following the similar idea as before, we consider the quantity G ( x, t ) := ( κ κ − α − ( κ − κ ) (4.5)to deduce the pinching estimate of the principal curvatures of M t . We first derive the evolutionequation of G along the flow (4.1). Lemma 4.2. Along the flow (4.1) , the evolution of G satisfies ∂∂t G = α ( K − α − ˙ K ij ∇ i ∇ j G + 2 α ( K − α − ( a T + a T ) (4.6) at the spatial maximum point of G . Here T , T are defined as (2.19) , and the coefficients a , a are given by a ( α, κ , κ ) :=4 κ κ ( κ − κ ) · α + ( κ − κ )( κ − κ κ + 5 κ − κ κ + 2 κ κ − κ ) · α + ( κ − κ + 4 κ − κ κ − κ κ − κ ) (4.7) and a ( α, κ , κ ) := a ( α, κ , κ ) .Proof. By a similar argument as in the proof of Lemma 3.2, the derivatives ˙ g , ˙ g can not vanishat the same time. We apply the general formula (2.29) to derive the coefficients in front of T : Z =2 α (2 α − κ − κ )( κ − κ κ − α − ( κ + κ − α ( α − κ − κ )( κ − κ κ − α − ( κ + κ − − α (2 α − κ − κ ) ( κ − κ κ − α − − α ( α − κ − κ ) ( κ − κ κ − α − − α ( κ + κ )( κ − κ ) κ ( κ κ − α − − α ( κ + κ )( κ − ( κ κ − α − + 4 α ( κ + κ ) κ κ ( κ − κ ) ( κ κ − α − − α κ κ ( κ − κ ) ( κ − κ κ − α − + 4 ακ ( κ − κ − κ − κ ) ( κ κ − α − − α κ κ ( κ − κ ) ( κ κ − α − . Finally, dividing Z by 2 α ( κ κ − α − and rearranging it as a quadratic polynomial of α , weobtain the desired expression (4.7). (cid:3) We now apply maximum principle to the evolution equation (4.6) to prove the monotonicityof max M t G ( x, t ) along the flow (4.1) with α ∈ [1 / , Theorem 4.3. Let M t , t ∈ [0 , T ) be a smooth solution of the flow (4.1) in H with K > ,where α ∈ [1 / , . Then max M t G ( x, t ) is monotone non-increasing in time.Proof. We need to show that the coefficients a i ( α, κ , κ ) are non-positive at the spatial maxi-mum point of G . Similar to the proof of Theorem 3.3, we denote C := { ( κ , κ ) ∈ R | κ κ > , κ > κ } . We assume κ > κ without loss of generality. Then the expression (4.7) says that both thecoefficients a i ( α, κ , κ ), i = 1 , α for each fixed ( κ , κ ) ∈ C . ONTRACTION OF SURFACES 19 We obviously have a (1 , κ , κ ) = − κ ( κ κ − ≤ ,a (1 , κ , κ ) = − κ ( κ κ − ≤ . We claim that a i (1 / , κ , κ ) ≤ , ∀ ( κ , κ ) ∈ C . Therefore, a i ( α, κ , κ ) ≤ κ , κ ) ∈ C provided that α ∈ [1 / , a (1 / , κ , κ ) as a (1 / , κ , κ ) = − 14 ( κ − κ ) (cid:0) κ − κ ) + 7 κ (cid:1) − κ ( κ − κ − κ ) − κ ( κ − = − (cid:0) κ − κ ) + 7 κ (cid:1) x − κ xy − κ y , where x = κ − κ and y = κ − 1. For ( κ , κ ) ∈ C , a (1 / , κ , κ ) is a strictly concave quadraticpolynomial of x, y , with discriminant∆ = 25 κ − κ (cid:0) κ − κ ) + 7 κ (cid:1) = − κ − κ ( κ − κ ) < , which implies that a (1 / , κ , κ ) ≤ κ , κ ) ∈ C .On the other hand, to show that a (1 / , κ , κ ) ≤ κ , κ ) ∈ C , we introduce theauxiliary function h ( x ) := a (1 / , x, κ )= 14 (cid:18) − κ + ( − 16 + 9 κ ) x + 11 κ x + (15 − κ ) x − κ x − x (cid:19) and denote h ( k ) ( x ) the k -th derivatives of h ( x ). Then we have h (2) ( x ) = − (cid:0) − κ − x + 21 κ + 36 κ x + 30 x (cid:1) = − (cid:18) κ ( x − 1) + 25 x ( κ x − 1) + 20 x ( x − 1) + 10 x + 21 κ (cid:19) . Hence, we have h (2) ( κ ) ≤ κ > max { κ , /κ } ≥ κ ≥ κ , then κ ≥ h (1) ( κ ) = − ( κ − κ − ≤ h ( κ ) = − κ ( κ − ≤ κ ≤ κ , then 0 < κ ≤ h (1) (1 /κ ) = − κ (1 − κ )(5 − κ + 3 κ ) ≤ h (1 /κ ) = − κ (1 − κ ) (1 − κ + κ ) ≤ h ( κ ) = a (1 / , κ , κ ) ≤ κ , κ ) ∈ C .Finally, by maximum principle we conclude that max M t G ( x, t ) is monotone non-increasingalong the flow (4.1) with α ∈ [ , (cid:3) Corollary 4.4. Let M t , t ∈ [0 , T ) be a smooth solution to the flow (4.1) with positive scalarcurvature in H , with α ∈ [1 / , . Then there exists a constant C > depending only on M and α such that < C ≤ κ κ ≤ C (4.8) on M t for all t ∈ [0 , T ) .Proof. As in the proof of Corollary 3.4, κ κ + κ κ − κ κ − α − ( κ − κ ) · κ κ ( κ κ − α − ≤ max M G ( · , 0) 1( κ κ − α − , which is bounded from above by Proposition 4.1 and α ≥ / 2. The pinching estimate (4.8)follows immediately. (cid:3) Flow in H by powers of Gauss curvature In this section, we study the flow of surfaces in H by powers of Gauss curvature, i.e., ∂∂t X ( x, t ) = − K α ( x, t ) ν ( x, t ) , α > X ( x, 0) = X ( x ) . (5.1) Proposition 5.1. Let M t , t ∈ [0 , T ) be the smooth solution to the flow (5.1) in H . For anypower α > , if M has positive scalar curvature, then M t has positive scalar curvature for all t ∈ [0 , T ) . Moreover, we have the estimate K − ≥ min M ( K − (cid:18) − (2 α + 1) min M ( K − α + t (cid:19) − α +1 (5.2) on M t for t ∈ [0 , T ) .Proof. By the equation (2.4), the speed function F = K α of the flow (5.1) evolves by ∂∂t K α = αK α − ˙ K ij ∇ i ∇ j K α + αK α − ˙ K ij (( h ) ij − g ij )= αK α − ˙ K ij ∇ i ∇ j K α + αK α − H ( K − . (5.3)Since α > 0, the maximum principle applied to (5.3) implies that K > (cid:3) We define G ( x, t ) := ( κ κ ) α ( κ − κ ) ( κ κ − on M t . By Proposition 5.1, the function G is well-defined on M t for all t ∈ [0 , T ). Theorem 5.2. Let M t , t ∈ [0 , T ) be a smooth solution of the flow (5.1) in H with K > ,where α ∈ [1 / , . Then max M t G ( x, t ) is monotone non-increasing in time. ONTRACTION OF SURFACES 21 Proof. The evolution of G along the flow (5.1) satisfies ∂∂t G = αK α − ˙ K ij ∇ i ∇ j G + 2 α ( κ κ ) α − ( κ κ − ( a T + a T ) . (5.4)at the spatial maximum point of G . Here T , T are defined as (2.19), and the coefficients a , a are given by a ( α, κ , κ ) :=4 κ ( κ − κ ) ( κ κ − · α + ( κ − κ )( κ κ − (cid:2) κ (1 − κ ) − κ ( κ − κ ) + (5 κ − κ ) (cid:3) · α + ( κ − (cid:2) κ κ + κ (1 − κ ) + κ ( κ − κ ) + κ (3 κ − κ ) − κ (cid:3) , (5.5)and a ( α, κ , κ ) := a ( α, κ , κ ). This can be proved by a similar argument as in the proof ofLemma 3.2.To prove Theorem 5.2, we need to show a i are non-positive at the spatial maximum pointof G . Both the coefficients a i ( α, κ , κ ), i = 1 , α in the cone C = { ( κ , κ ) ∈ R | κ κ > , κ > κ } . We claim that for all ( κ , κ ) ∈ C , a i (1 / , κ , κ ) ≤ , a i (1 , κ , κ ) ≤ . Then, for all ( κ , κ ) ∈ C , a i ( α, κ , κ ) ≤ α ∈ [1 / , g i,α ( x ) := a i ( α, x, κ ), where ( κ , κ ) ∈ C . It suffices to showthat g i, / ( κ ) ≤ , g i, ( κ ) ≤ . Firstly, we show that g i, ( κ ) ≤ i = 1 , 2. We have g (2)1 , ( x ) = − − κ − κ + x + κ x + 2 κ x )= − (cid:2) ( κ + 1)( x − κ ) + 2 κ ( κ x − (cid:3) , and g (3)2 , ( x ) = − − − κ + 12 κ x + 8 κ x + 10 x )= − (cid:2) x − κ ) + 5( κ x − 1) + x + 8 κ x + 7 κ x (cid:3) . Hence, we have g (2)1 , ( κ ) ≤ g (3)2 , ( κ ) ≤ κ > max { κ , /κ } ≥ κ ≥ κ , then κ ≥ g (1)1 , ( κ ) = − κ ( κ − ≤ g , ( κ ) = − κ ( κ − ≤ g (2)2 , ( κ ) = − κ ( − κ + 6 κ ) ≤ g (1)2 , ( κ ) = − κ ( κ − ≤ g , ( κ ) = − κ ( κ − ≤ κ ≤ κ , then 0 < κ ≤ g (1)1 , (1 /κ ) = − κ ( κ − (1 + κ ) ≤ g , (1 /κ ) = − κ ( κ − (1 + κ ) ≤ and g (2)2 , (1 /κ ) = − κ (1 + κ )(2 − κ )(5 − κ ) ≤ g (1)2 , (1 /κ ) = − κ (1 − κ )(5 + 2 κ + κ ) ≤ g , (1 /κ ) = − κ (1 − κ ) (1 + κ ) ≤ g i, ( κ ) ≤ g i, / ( κ ) ≤ 0. We have g (3)1 , / ( x ) = − 32 (5 − κ + 7 κ + 12 κ x )= − (cid:2) κ ( x − κ ) + (2 κ − + 3 κ + 4 (cid:3) ≤ , and g (3)2 , / ( x ) = − (cid:2) − κ x + 25 x + 6 κ x ( − x ) + 6 κ ( − x ) (cid:3) ≤ , Hence, we have g (3)1 , / ( κ ) ≤ g (3)2 , / ( κ ) ≤ κ > max { κ , /κ } ≥ κ ≥ κ , then κ ≥ g (2)1 , / ( κ ) = − κ ( κ − κ − ≤ g (1)1 , / ( κ ) = − κ ( κ − ≤ g , / ( κ ) = − κ ( κ − ≤ g (2)2 , / ( κ ) = − κ ( − − κ + 147 κ ) ≤ g (1)2 , / ( κ ) = − κ ( κ − − κ ) ≤ g , / ( κ ) = − κ ( κ − ≤ κ ≤ κ , then 0 < κ ≤ g (2)1 , / (1 /κ ) = − κ (1 − κ )(11 − κ − κ ) ≤ g (1)1 , / (1 /κ ) = − κ (1 − κ ) (3 − κ + κ )(3 + 2 κ + κ ) ≤ g , / (1 /κ ) = − κ (1 − κ ) (1 + κ ) ≤ g (2)2 , / (1 /κ ) = − κ (cid:0) − κ + 27 κ − κ (cid:1) ≤ g (1)2 , / (1 /κ ) = − κ (1 − κ )(43 − κ + 29 κ − κ ) ≤ g , / (1 /κ ) = − κ (1 − κ ) (1 + κ ) ≤ ONTRACTION OF SURFACES 23 In both cases, we have g i, / ( κ ) ≤ a , a in (5.4) are non-positive at the spatial maximum point of G .Applying the maximum principle, we conclude that max M t G ( x, t ) is monotone non-increasingalong the flow (5.1) with α ∈ [ , (cid:3) Applying similar argument as in Corollary 4.4, we have Corollary 5.3. Let M t , t ∈ [0 , T ) be a smooth solution of the flow (5.1) with K > in H ,where α ∈ [1 / , . There exists a constant C > depending only on M and α such that < C ≤ κ κ ≤ C on M t for all t ∈ [0 , T ) . Contracting flows in the sphere In this section, we study the flows for strictly convex surfaces in the sphere S . We will provethe curvature pinching estimates along the flow.6.1. Flow by powers of mean curvature. In this subsection, we study the flow of closedsurfaces in the sphere by powers of mean curvature. Let X : M → S be a smooth, closed andstrictly convex surface in the sphere. We consider a family of closed surfaces M t = X ( M, t ) in S contracting with normal velocity F = H α in (1.1), i.e., ∂∂t X ( x, t ) = − H α ( x, t ) ν ( x, t ) , α > X ( x, 0) = X ( x ) . (6.1)We first show that the strict convexity of M t is preserved along the flow (6.1). Proposition 6.1. Let M t , t ∈ [0 , T ) be the smooth solution to the flow (6.1) in S . For anypower α > , if M is strictly convex, then M t is strictly convex for all t ∈ [0 , T ) . Moreover, thespatial minimum of the mean curvature in non-decreasing in time and satisfies the estimate H ( x, t ) ≥ min M H ( · , (cid:18) − α + 1 n (min M H ( · , α +1 t (cid:19) − α +1 . (6.2) for all t ∈ [0 , T ) . This implies an upper bound for the maximum existence time T : T ≤ nα + 1 (min M H ( · , − ( α +1) Proof. To show the strict convexity of M t for t > 0, we prove that the Gauss curvature K > G = K in (2.10). Similar with the Euclidean case inProposition 3.1, the zero order term Q of the evolution equation of K satisfies Q =(1 − α ) KH α +1 + 2 αK ( | A | − H α − + ( α + 1) H α +1 =( K + 1)( H + ( κ − κ ) α ) H α − ≥ . The gradient term Q is the same as in R . Hence, by (3.5) we have Q ≥ K . By the maximum principle, we have min M t K ≥ min M K > 0, and hence M t is strictly convex for t ∈ [0 , T ). By the equation (2.4), the speed function F = H α of the flow (6.1) evolves by ∂∂t H α = αH α − ∆ H α + αH α − ( | A | + 2) . which is equivalent to ∂∂t H =∆ H α + H α ( | A | + 2) . This implies that ddt min M t H ≥ H α +2 /n. Then the estimate (6.2) follows from the maximum principle. (cid:3) As we mentioned in § 1, we consider the function G ( x, t ) = g ( κ ) = ( κ − κ ) ( κ + κ ) α ( κ κ ) (6.3)to derive the curvature pinching estimate of the flow in the sphere S . This is the same one usedin [28] in the Euclidean case. Theorem 6.2. Let M t , t ∈ [0 , T ) be a smooth strictly convex solution of the flow (6.1) in S ,where α ∈ [1 , . Then max M t G ( x, t ) is monotone non-increasing in time.Proof. We apply (2.6) to calculate the evolution equation of G along the flow (6.1). We have ∂∂t G = ˙ F ij ∇ i ∇ j G + Q + Q , (6.4)at the spatial maximum point of G , where Q , Q denotes the gradient term and zero order termin the evolution of G . By the definition (6.3), the derivatives of G are given by˙ g = 2 κ − κ − ( κ − κ ) H α − ( κ ( κ + κ ) + κ ( κ − κ ) α ) , ˙ g = 2 κ − κ − ( κ − κ ) H α − ( κ ( κ + κ ) + κ ( κ − κ ) α ) . A direct calculation then gives˙ G ij ( h ) ij =2 αH − G ( κ + κ ) , ˙ G ij h ij = 2( α − G, ˙ G ij g ij = − H − K − G ( H − αK ) . The zero-order term Q of (6.4) for G can be computed as follows: Q = − ˙ G ij ( h ) ij ˙ F kl h kl + ˙ G ij h ij ˙ F kl ( h ) kl + F ˙ G ij ( h ) ij + ( ˙ G ij g ij ˙ F kl h kl − ˙ G ij h ij ˙ F kl g kl + F ˙ G ij g ij )= − H α − K − G ( H + α ( κ − κ ) ) ≤ . Since the function G is the same one used in [28] for the flow in R by powers of mean curvature,the gradient term Q in the evolution equation (6.4) of G is the same as in the Euclidean case.Therefore by the argument as in [28, Lemma A.2], Q ≤ G provided that the power α ∈ [1 , M t G ( x, t ) is monotone non-increasing along the flow (6.1) with α ∈ [1 , (cid:3) ONTRACTION OF SURFACES 25 Corollary 6.3. Let M t , t ∈ [0 , T ) be a smooth strictly convex solution of H α -flow in S , where α ∈ [1 , . There exists a constant C > depending only on the initial surface M and α suchthat < C ≤ κ κ ≤ C, (6.5) on M t for all t ∈ [0 , T ) .Proof. κ κ + κ κ − κ − κ ) κ κ = ( κ − κ ) H α ( κ κ ) · κ κ H α ≤ max M G ( · , H α − , which is bounded from above by (6.2) and α ≥ 1. Hence, the estimate (6.5) follows immediately. (cid:3) Flow by powers of Gauss curvature. In this subsection, we study the flow for surfacesin the sphere by powers of Gauss curvature, i.e., ∂∂t X ( x, t ) = − K α ( x, t ) ν ( x, t ) , α > X ( x, 0) = X ( x ) . (6.6) Proposition 6.4. Let M t , t ∈ [0 , T ) be the smooth solution to the flow (6.6) in S . For anypower α > , if M is strictly convex, then M t is strictly convex for all t ∈ [0 , T ) . Moreover,wehave the estimate K ≥ min M K (cid:18) − (2 α + 1) min M K α + t (cid:19) − α +1 (6.7) on M t for t ∈ [0 , T ) .Proof. This follows from a similar argument as in Proposition 5.1. By the equation (2.4), thespeed function F = K α of the flow (6.6) evolves by ∂∂t K α = αK α − ˙ K ij ∇ i ∇ j K α + αK α − H ( K + 1) . (6.8)Since α > K > ddt min M t K ≥ K α + . Then the estimate (6.7) follows by integrating the above inequality. (cid:3) We consider the following function G ( x, t ) = g ( κ ) = ( κ − κ ) ( κ κ ) − α . (6.9) Theorem 6.5. Let M t , t ∈ [0 , T ) be a smooth strictly convex solution of the flow (6.6) in S ,where α ∈ [1 / , . Then max M t G ( x, t ) is monotone non-increasing in time. Proof. Since F = f ( κ ) = ( κ κ ) α and G is defined as in (6.9), the derivatives of F and G aregiven by ˙ f = ακ − ( κ κ ) α , ˙ f = ακ − ( κ κ ) α , ˙ g = 2 κ κ ( κ − κ )( κ κ ) α ( κ + ( κ − κ ) α ) , ˙ g = 2 κ κ ( κ − κ )( κ κ ) α ( κ + ( κ − κ ) α ) . (6.10)We apply (2.6) to calculate the evolution equation of G along the flow (6.6), i.e., at the spatialmaximum point of G we have: ∂∂t G = ˙ F ij ∇ i ∇ j G + Q + Q , (6.11)where Q , Q denotes the gradient term and zero order term. By (2.6) and (6.10), the zero-orderterm of (6.11) for G satisfies Q = − (2 α − f ( ˙ g κ + ˙ g κ ) + 2(2 α − g ( ˙ f κ + ˙ f κ )+ (2 α + 1) f ( ˙ g + ˙ g ) − α − g ( ˙ f + ˙ f )= − HK − f g ≤ . (6.12)Now we apply (2.20) to calculate the gradient term Q at the maximum point ( p, t ), Q = (cid:18) ( ˙ g ¨ f − ˙ f ¨ g ) γ − 2( ˙ g ¨ f − ˙ f ¨ g ) βγ + (cid:0) ˙ g ¨ f − ˙ f ¨ g + 2 ˙ g ˙ f − ˙ g ˙ f κ − κ (cid:1) β (cid:19) T + (cid:18) ( ˙ g ¨ f − ˙ f ¨ g ) β − 2( ˙ g ¨ f − ˙ f ¨ g ) βγ (6.13)+ (cid:0) ˙ g ¨ f − ˙ f ¨ g + 2 ˙ g ˙ f − ˙ g ˙ f κ − κ (cid:1) γ (cid:19) T , where β and γ in (6.13) are given by β := κ ( κ + ( κ − κ ) α ) > , γ := κ ( − κ + ( κ − κ ) α ) < . Note that the derivation of (2.20) didn’t use any ambient curvatures, and so can be appliedhere. Taking a further derivative to the equations (6.10), we have¨ f = α ( α − κ − K α , ¨ f = α ( α − κ − K α , ¨ f = α K α − , ¨ g =2 K α − κ − (cid:2) κ ( − κ + 3 κ ) − ( κ − κ )( κ − κ ) α + 2( κ − κ ) α (cid:3) , ¨ g =2 K α − κ − (cid:2) κ ( − κ + 3 κ ) − ( κ − κ )( κ − κ ) α + 2( κ − κ ) α (cid:3) , ¨ g =2 K α − (cid:2) − κ κ − κ − κ ) α + 2( κ − κ ) α (cid:3) . (6.14)We only calculate the coefficient (denoted by Z ) in front of T in the equation (6.13) as in (2.21).Using (6.10) and (6.14), we obtain the expression for Z as follows: Z =2 αK α − κ − (cid:18) κ ( α − 1) + κ ( α − α − κ κ ( α − α + 1) + κ κ ( − α − α ) (cid:19) . ONTRACTION OF SURFACES 27 For α ∈ [ , Z ≤ 0. This means that the gradient term Q in theevolution equation (6.11) is non-positive at the spatial maximum point of G . Applying themaximum principle, we conclude that max M t G ( x, t ) is monotone non-increasing along the flow(6.6) with α ∈ [ , (cid:3) Remark . The gradient term (6.13) is same as the Euclidean case in [9]. The computation in[9] is carried out using the Gauss map parametrization of the flow: The flow (6.6) in Euclideanspace R is equivalent to a scalar parabolic equation on the sphere S for the support function ofthe evolving surfaces. Here we prove our estimate using the calculation on the evolving surfacesdirectly. Corollary 6.7. Let M t , t ∈ [0 , T ) be a smooth strictly convex solution of the flow (6.6) in S ,where α ∈ [1 / , . There exists a constant C > depending only on M and α such that < C ≤ κ κ ≤ C (6.15) on M t for all t ∈ [0 , T ) .Proof. As before, we have κ κ + κ κ − κ − κ ) ( κ κ ) − α · κ κ ) α − ≤ max M G ( · , · κ κ ) α − , which is bounded from above by the estimate (6.7) and α ≥ / 2. Then the estimate (6.15)follows. (cid:3) Convergence In this section, we discuss the convergence of the solution to a point and of the rescaledsolution to a sphere. We only give the details for the flow in H by powers of mean curvature,since the proof is similar for the remaining flows.7.1. Contraction to a point.Proposition 7.1. The evolving surfaces M t of the flow (3.1) remain smooth until they contractto a point as t → T .Proof. Let ρ + ( t ) and ρ − ( t ) be the outer radius and inner radius of the domain Ω t enclosed by M t , defined by ρ + ( t ) = inf { ρ : Ω t ⊂ B ρ ( p ) for some p ∈ H } ρ − ( t ) = sup { ρ : B ρ ( p ) ⊂ Ω t for some p ∈ H } . By the pinching estimate (3.12), we can apply a similar argument in [21, § 6] (see also [12, § ρ + ( t ) ≤ Cρ − ( t ) , for t ∈ [ t , T ) , (7.1)where t is sufficiently close to T . This makes sense because it holds for surfaces with pinchedcurvatures in Euclidean space (by Andrews [2]), and small surface in hyperbolic space is com-parable to its analogue in Euclidean space in the conformal flat coordinate system.The technique of Tso [29] can be used to show that the mean curvature remains boundedas long as the flow encloses a non-vanishing volume: Assume that there exits a geodesic ball B ρ ( x ) ⊂ Ω t for t ∈ [0 , t ], where t ∈ [ t , T ). Since M t is strictly convex, we can write M t = graph u ( · , t ) as graphs in polar coordinates centered at x . Then u ≥ ρ for all t ∈ [0 , t ]. By the comparison principle, the later surface is contained in the earlier one, then we have anupper bound on u ≤ R for some constant R > M . Denote by ∂ r = ∂ r x thegradient vector at x ∈ M t along the geodesic from x to x . The support function of M t withrespect to x is defined by χ ( x, t ) = sinh u ( x, t ) h ∂ r , ν i . Due to the strict convexity of M t and ρ ≤ u ≤ R , we have (see [20]) v = 1 + sinh − u | Du | g S ≤ e κ ( u max − u min ) where ¯ κ is an upper bound of the principal curvatures of the slices intersecting M t . Then¯ κ ≤ coth u min and v ≤ e coth u min ( u max − u min ) ≤ e coth ρ ( R − ρ ) . Hence h ∂ r , ν i = v − ≥ e − coth ρ ( R − ρ ) =: 2 ǫ , where ǫ > ρ and M . Then because u ( x, t ) ≥ ρ , there holds χ > ǫ sinh ρ, ∀ t ∈ [0 , t ]and the function ϕ = H α χ − ǫ sinh ρ is well defined on M t for all t ∈ [0 , t ]. Recall that by (2.4) the mean curvature satisfies theevolution equation ∂∂t H α = αH α − ∆ H α + αH α − ( | A | − . (7.2)and the support function satisfies (see [14, § ∂∂t χ = αH α − ∆ χ + αH α − | A | χ − (1 + α ) H α cosh u ( x ) . The function ϕ satisfies the evolution equation ∂∂t ϕ = αH α − (cid:18) ∆ ϕ + 2 χ − ǫ sinh ρ g ij ∇ i χ ∇ j ϕ (cid:19) + (cid:18) (1 + α ) cosh u ( x ) − ǫ sinh ρα | A | H (cid:19) ϕ − αH α − ϕ ≤ αH α − (cid:18) ∆ ϕ + 2 χ − ǫ sinh ρ g ij ∇ i χ ∇ j ϕ (cid:19) + (cid:16) (1 + α ) cosh u ( x ) − α ǫ sinh ρ ) α +1 α ϕ /α (cid:17) ϕ , (7.3)where we used | A | ≥ H / χ − ǫ sinh ρ > ǫ sinh ρ . Applying maximum principleto (7.3) gives the upper bound on ϕ . This together with the upper bound on χ implies that H is bounded from above by a constant depending on ρ, α, M .On the other hand, by Proposition 3.1 we have H ≥ √ K > 2. This together with the upperbound on mean curvature and the pinching estimate (3.12) implies that all the principal curva-tures of M t are bounded above and below by positive constants. In particular, the coefficients˙ F ij = αH α − g ij in the second order part of the problem have eigenvalues bounded above andbelow by positive constants, and then the flow remains to be uniformly parabolic. By applyingthe H¨older estimate of the second derivatives of uniformly parabolic equation of Andrews [6],and standard Schauder theory, we can derive the higher regularity estimates of the solution tothe flow. It follows that the solution can be extended past time t . This means that the smoothsolution of the flow (3.1) exists as long as the evolving domain encloses a non-vanishing volume.Therefore the inner radius ρ − ( t ) → t → T . The estimate (7.1) then says that the outer ONTRACTION OF SURFACES 29 radius converges to zero as t → T as well. In other words, the flow remains smooth until itcontracts to a point as t → T . (cid:3) Convergence of the rescaled solution. To study the asymptotical behavior of the flow,we consider the rescaling of the solution using a similar procedure in [21] for the contractingflow in the sphere. If the initial surface is a geodesic sphere in H , the evolving surfaces M t areall geodesic spheres with the same center and radius Θ = Θ( t, T ) satisfying ddt Θ = − α coth α Θ . The spherical solution shrinks to a point in finite time. This also implies that the maximumexistence time of the flow (3.1) is finite. Let T be the maximum existence time as in Lemma7.1. We define the rescaled mean curvature by ˜ H = Θ( t, T ) H . Since we only care about theasymptotical behavior of the flow, we may focus on the flow in the time interval [ t , T ), where t is the time such that the pinching estimate (7.1) holds in [ t , T ). Lemma 7.2. There exists a uniform constant C such that ˜ H ≤ C, ∀ t ∈ [ t , T ) . (7.4) Proof. For any t ∈ ( t , T ), let B ρ − ( t ) ( x ) be an inball of M t . Write M t = graph u ( x, t ) asgeodesic radial graphs with respect to the point x for all t ∈ [ t , t ]. Then ρ − ( t ) ≤ u ( t ) ≤ u ( t ) ≤ R . Since M t is strictly convex, the minimum of χ is achieved at the minimum point of u , which implies that χ ≥ sinh ρ − ( t ) for all t ∈ [ t , t ]. Applying maximum principle to (7.3),we have ϕ ( t ) ≤ max (cid:26) ϕ ( t ) (cid:18) α + 14 ( 12 sinh ρ − ( t )) α +1 α ϕ ( t ) α +1 α ( t − t ) (cid:19) − α α , (cid:18) α ) α cosh R (cid:19) α ( 12 sinh ρ − ( t )) − α − (cid:27) . Choosing t sufficiently close to T , we can make ρ − ( t ) small enough such that ϕ ( t ) ≤ (cid:18) α ) α cosh R (cid:19) α ( 12 sinh ρ − ( t )) − α − . Then H α ( t )(sinh ρ − ( t )) α = ϕ ( t )( χ − 12 sinh ρ − ( t ))(sinh ρ − ( t )) α ≤ (cid:18) α ) α cosh R (cid:19) α ( 12 sinh ρ − ( t )) − ( χ − 12 sinh ρ − ( t )) ≤ (cid:18) α ) α cosh R (cid:19) α ( 12 sinh ρ − ( t )) − (sinh 2 ρ + ( t ) − 12 sinh ρ − ( t )) ≤ C, where in the last inequality we used the estimate (7.1). The estimate (7.4) follows becauseΘ( t , T ) is comparable with sinh ρ − ( t ) for t sufficiently close to T . Indeed, by the comparisonprinciple the spherical solution of radius Θ( t, T ) must intersect M t for t ∈ [0 , T ). This impliesthat inf M t u ( · , t ) ≤ Θ( t, T ) ≤ sup M t u ( · , t ). Combining this with the pinching estimate (7.1), wehave ρ − ( t ) ≤ Θ( t , T ) ≤ ρ + ( t ) ≤ Cρ − ( t ). For t sufficiently close to T , this is equivalentto that Θ( t , T ) is comparable with sinh ρ − ( t ). (cid:3) Let us define a new time parameter τ = − log Θ. Then dτdt = − ddt Θ = 2 α Θ − coth α Θ . From (7.2) we obtain ∂∂τ ˜ H = ∂∂t (Θ H ) dtdτ = 2 − α tanh α (Θ)Θ ∂∂t H − ˜ H =2 − α tanh α (Θ)Θ (cid:0) ∆ H α + H α ( | A | − (cid:1) − ˜ H =2 − α Θ − α tanh α (Θ)Θ ∆ ˜ H α + 2 − α Θ − α tanh α (Θ) ˜ H α ( | ˜ A | − ) − ˜ H, (7.5)where ˜ A = Θ A denotes the rescaled second fundamental form. The upper bound (7.4) togetherwith the pinching estimate (3.12) implies that the rescaled principal curvatures ˜ κ i = Θ κ i areuniformly bounded from above.Let t ∈ [ t , T ) be arbitrary and let t > t such thatΘ( t , T ) = 2Θ( t , T ) . Then τ i = − log Θ( t i , T ) satisfies τ = τ + log 2. Introduce polar coordinates with respect tothe center of an inball of Ω t and write M t as graphs of u ( x, t ) for t ∈ [ t , t ]. Then the pinchingestimate (7.1) implies C − Θ( t, T ) ≤ u ( x, t ) ≤ C Θ( t, T ) , ∀ t ∈ [ t , t ] (7.6)and u max ( t ) ≤ C u min ( t ), for all t ∈ [ t , t ]. Since M t is strictly convex, this implies that v = 1 + sinh − u | Du | g S (7.7)is uniformly bounded in [ t , t ] × S . Lemma 7.3. The rescaled mean curvature ˜ H satisfies the following porous medium equation ofthe form ∂∂τ ˜ H = ¯ ∇ i ( a ij ¯ ∇ j ˜ H α ) + b i ∂ i ˜ H α + c ˜ H (7.8) in the cylinder Q ( τ , τ ) = [ τ , τ ] × S with uniformly bounded coefficients b i and c , and C − ≤ ( a ij ) ≤ C (7.9) independent of τ i . Here τ i = − log Θ( t i , T ) satisfies τ = τ + log 2 , ¯ ∇ denotes the covariantderivative on S with respect to the standard metric g S = ( σ ij ) .Proof. By (7.5), the evolution of ˜ H satisfies ∂∂τ ˜ H =2 − α Θ − α tanh α (Θ)Θ ∇ i ( g ij ∇ j ˜ H α )+ 2 − α Θ − α tanh α (Θ) ˜ H α ( | ˜ A | − ) − ˜ H =2 − α Θ − α tanh α (Θ) (cid:16) ¯ ∇ i (Θ g ij ¯ ∇ j ˜ H α ) + Θ g ij (Γ kij − ¯Γ kij ) ∂ k ˜ H α (cid:17) + 2 − α Θ − α tanh α (Θ) ˜ H α ( | ˜ A | − ) − ˜ H, (7.10)where g ij is the inverse of the metric g ij = u i u j + sinh uσ ij = sinh u ( ϕ i ϕ j + σ ij ), Γ kij and ¯Γ kij are Christoffel symbols of the metric g ij and σ ij respectively. Here ϕ is defined such that ϕ i = u i sinh u . ONTRACTION OF SURFACES 31 By the estimate (7.7) on v , ϕ i is uniformly bounded. Hence g ij Θ ≈ sinh − u Θ is uniformlybounded from above and from below by positive constants in view of (7.6). Since Θ is small inthe interval [ t , T ), we also have uniform bound on Θ − α tanh α Θ. This gives the estimate (7.9)on the coefficients a ij . To estimate the bound on b i , we notice thatΓ kij − ¯Γ kij = 12 g kq ( ¯ ∇ i g jq + ¯ ∇ j g iq − ¯ ∇ q g ij ) , which depends on the first and second derivatives of ϕ . Recall that the Weingarten matrix ofthe graph M t = graph u ( x, t ) is given by (see [20]) h ji = − v sinh u ( σ ik − ϕ i ϕ k v ) ϕ jk + v − coth uδ ji . (7.11)Since the rescaled Weingarten matrix ˜ h ji = Θ h ji and ϕ i are uniformly bounded, The equation(7.11) gives the upper bound on ϕ ij , the second derivatives of ϕ . Then Γ kij − ¯Γ kij is uniformlybounded. Finally, the bound on the zero order term c follows from upper bound on the rescaledprincipal curvatures. This completes the proof of the lemma. (cid:3) We can apply [18, Theorem 1.2] to (7.8) to deduce the H¨older continuity estimate for ˜ H on theregion Q ( τ , τ ), with the constant depending on R Q ( τ ,τ ) | ¯ ∇ ˜ H α | dµ S dτ . To bound this term,by (7.2) and integration by parts, we have ddt Z M t H α +1 dµ t = ( α + 1) Z M t H α (cid:0) ∆ H α + H α ( | A | − (cid:1) dµ t − Z M t H α +1) dµ t ≤ α Z M t H α +1) dµ t − ( α + 1) Z M t |∇ H α | dµ t , where we used | A | ≤ H since each M t is strictly convex. Equivalently, ddτ Z M t H α +1 dµ t = ddt Z M t H α +1 dµ t · dtdτ ≤ − α Θ tanh α Θ (cid:18) α Z M t H α +1) dµ t − ( α + 1) Z M t |∇ H α | dµ t (cid:19) , where τ = − log Θ( t, T ). Multiplying the two sides of the above inequality by 2 α Θ α − tanh − α Θ / ( α + 1)and integrating from τ to τ , we obtainΘ α Z τ τ Z M t |∇ H α | dµdτ ≤ αα + 1 Θ α Z τ τ Z M t H α +1) dµdτ − α α + 1 Θ α − tanh − α Θ Z M t H α +1 dµ (cid:12)(cid:12)(cid:12)(cid:12) τ τ . (7.12)Since g ij ≈ Θ σ ij , the left hand side of (7.12) is comparable with Z τ τ Z S | ¯ ∇ ˜ H α | g S dµ S dτ, while the right hand side of (7.12) is bounded uniformly using the estimate (7.4). The requiredbound on R Q ( τ ,τ ) | ¯ ∇ ˜ H α | dµ S dτ follows. Thus, applying Theorem 1.2 in [18], we obtain thatfor any ( x, τ ) ∈ S × [ τ + log 2 , ∞ ), there exist a universal constant δ > γ < γ -H¨older norm of ˜ H on the space-time neighborhood B δ ( x ) × [ τ − δ, τ + δ ] is uniformlybounded, where B δ ( x ) denotes a geodesic ball of radius δ centered at x in S .Let p ∈ H be the point the flow surfaces are shrinking to. Introduce geodesic polar co-ordinates around p and write M t as graphs of u ( x, t ), ( x, t ) ∈ S × [ t , T ). We consider the rescaled function ˜ u ( x, τ ) = u ( x, t )Θ − ( t, T ) on ( x, τ ) ∈ S × [ τ , ∞ ). Note that the rescaledprincipal curvatures ˜ κ i = Θ κ i are not the principal curvatures of the graph of ˜ u , though theyare related. By the proof of Lemma 7.3, the C -norm of ˜ u is uniformly bounded. Thus for anysequence of time τ j , there exists a subsequence (still denoted by τ j ) such that ˜ u ( · , τ j ) convergesin C ,γ to a limit function ˜ u ∞ for any γ < 1. At each time τ j , let p j ∈ S be a point such that˜ u max ( τ j ) = ˜ u ( p j , τ j ). Then ˜ κ i ( p j , τ j ) = κ i ( p j , t j )Θ( t j , T ) ≥ coth u ( p j , t j )Θ( t j , T ) ≥ C > 0, wherewe recall that τ j and t j are related by τ j = − log Θ( t j , T ). This implies that ˜ H ( p j , τ j ) ≥ C > H implies that ˜ H can not decrease too fast in the sense that ˜ H ≥ C in B δ ( p j ) × [ τ j − δ, τ j + δ ]. The rescaled function ˜ u ( x, τ ) now satisfies the uniformly parabolicequation ∂∂τ ˜ u = − − α tanh α Θ vH α + ˜ u = − − α Θ − α tanh α Θ v ˜ H α + ˜ u (7.13)in B δ ( p j ) × [ τ j − δ, τ j + δ ], where v is the function defined in (7.7). By the H¨older estimate [6]and Schauder estimate, we obtain uniform C ∞ estimate for the rescaled function ˜ u in B δ/ ( p j ) × [ τ j − δ/ , τ j + δ/ S is compact, there exists a point p ∞ ∈ S such that afterpassing to a subsequence we have p j → p ∞ . The above estimate implies that ˜ u ( x, τ j ) convergesto ˜ u ∞ in C ∞ for all x ∈ B δ/ ( p ∞ ).By Theorem 3.3, κ κ + κ κ − κ + κ ) α ( κ − κ ) ( κ κ − ( κ κ − κ κ ( κ + κ ) α ≤ CH − α ) = C ˜ H − α ) Θ α − which converges to zero as τ j → ∞ in B δ/ ( p ∞ ), because α > H ( x, τ j ) is bounded in B δ ( p j ). In other words,1 ≤ κ κ ≤ C Θ ( α − = 1 + Ce − ( α − τ j → τ j → ∞ (7.14)in B δ/ ( p ∞ ). By the inequality |∇ H | ≤ |∇ A | / § M t , we have |∇ A | ≤ |∇ ˚ A | ≤ | ˚ A ||∇ ˚ A | . We deduce that |∇ ˜ A | = Θ g ij (Θ h lk ; i )(Θ h kl ; j ) = Θ |∇ A | ≤ | ˚˜ A ||∇ ˚˜ A | ≤ Ce − ( α − τ j converges to zero in B δ/ ( p ∞ ) as τ j → ∞ , where we used the fact that |∇ ˚˜ A | is bounded due tothe regularity estimate of ˜ u = u Θ − . This implies that( ˜ H max − ˜ H min ) | B δ/ ( p ∞ ) ≤ Θ |∇ A | diam( M t ∩ graph u ( · , t j ) | B δ/ ( p ∞ ) )= |∇ ˜ A | Θ − diam( M t ∩ graph u ( · , t j ) | B δ/ ( p ∞ ) ) ≤ Ce − ( α − τ j , where τ j = − log Θ( t j , T ). Therefore, ˜ H ( x, τ j ) becomes arbitrary close to the value ˜ H ( p j , τ j ) in B δ/ ( p ∞ ). Using the H¨older continuity of ˜ H and repeating the above argument, we can extendthe region where ˜ u ( x, τ j ) converges in C ∞ to ˜ u ∞ to a larger one, say B δ ( p ∞ ). After a finitenumber of iterations, we deduce that ˜ u ( · , τ j ) converges in C ∞ to ˜ u ∞ on S . The above argumentcan be applied to any sequence τ j , we conclude that the whole family ˜ u ( · , τ ) satisfy uniform C ∞ ONTRACTION OF SURFACES 33 estimates on S , and converge to the same limit function ˜ u ∞ smoothly as τ → ∞ . Here theconvergence to the same limit function follows from the evolution equation (7.13) and Cauchycriterion. The interpolation inequality then implies that |∇ ˜ A | ( · , τ ) ≤ Ce − ( α − τ converges tozero as τ → ∞ on S . By a similar argument as in [21, § u → τ → ∞ .The smooth convergence of ˜ u to 1 again follows from the interpolation inequality. Thus, wecomplete the proof of Theorem 1.1 for the flow by powers of mean curvature. Proof of Theorem 1.1. We have given a detailed proof for the case (i). For case (ii), the flow bypowers of scalar curvature. A similar argument as in case (i) implies that the scalar curvature R = 2( K − 1) blows up as the final time T is approached. If α = 1 / 2, by the pinching estimate(4.8) we can check that the evolution equation of rescaled speed function Θ( t, T ) √ K − t, T ) √ K − t, T ) denotes the spherical solution to (4.1) with the same maximum existencetime T . The convergence of the rescaled solutions can be proved using a similar procedure as in[10]. For 1 / < α ≤ 1, the evolution equaiton of the rescaled speed function is a porous mediumtype equation as in Lemma 7.3. Then the argument given in the proof of case (i) can be adaptedto complete the proof. The proof for case (iii) is similar. (cid:3) Proof of Theorem 1.2. The case (i) with α = 1 and case (ii) with α = 1 / α = 1 was proved by McCoy [26]. For the remainingcases, the evolution equation of the rescaled speed function can be written as a porous mediumtype equation as in Lemma 7.3. The argument there can be adapted to complete the proof. (cid:3) Remark . The idea of applying the H¨older estimate of DiBenedetto and Friedman [18] forporous medium type equation to derive the regularity estimate for curvature flows in Euclideanspace has previously been used in [19, 28]. References [1] R. Alessandroni and C. Sinestrari, Evolution of hypersurfaces by powers of the scalar curvature , Ann. ScuolaNorm. Sup. Pisa Cl, Sci. (5) (2010), no. 3, 541-571.[2] B. Andrews, Contraction of convex hypersurfaces in Euclidean space , Calc. Var. Partial Differential Equations, (1994), no. 2, 151–171.[3] B. Andrews, Contraction of convex hypersurfaces in Riemannian spaces , J. Differential Geom. (1994), no.2,407–431.[4] B. Andrews, Gauss curvature flow: The fate of the rolling stones , Invent. Math. (1999), no.1, 151-161.[5] B. Andrews, Positively curved surfaces in the three-sphere , Proceedings of the ICM, Higher Ed. Press, Beijing2002, vol. , 221-230.[6] B. Andrews, Fully nonliear parabolic equations in two space variables , preprint 2004, arXiv:0402235.[7] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions , J. reine angew. Math. (2007), 17-33.[8] B. Andrews, Moving surfaces by non-concave curvature functions , Calc. Var. Partial Differential Equations (2010), no. 3-4, 649-657.[9] B. Andrews and X. Chen, Surfaces moving by powers of Gauss curvature , Pure Appl. Math. Q. (2012), no.4,825-834.[10] B. Andrews and X. Chen, Curvature flow in hyperbolic spaces , J. reine angew Math. (2017). 29-49.[11] B. Andrews, P. Guan and L. Ni, Flow by powers of the Gauss curvature , Adv. Math., , 174–201, 2016.[12] B. Andrews, Y. Hu and H. Li, Harmonic mean curvature flow and geometric inequalities , arXiv:1903.05903.[13] B. Andrews and J. McCoy, Convex hypersurfaces with pinched principal curvatures and flow of convex hy-persurfaces by high powers of curvature , Trans. Amer. Math. Soc. (2012), no. 7, 3427–3447.[14] B. Andrews and Y. Wei, Quermassintegral preserving curvature flow in hyperbolic space , Geom. Funct. Anal., (2018), no.5, 1183–1208.[15] S. Brendle, K. Choi and P. Daskalopoulos, Asymptotic behavior of flows by powers of the Gaussian curvature ,Acta. Math (2017), 1-16. [16] B. Chow, Deforming convex hypersurfaces by the n th root of the Gaussian curvature , J. Differential Geom. (1985), no.1, 117-138.[17] B. Chow, Deforming covnex hypersurfaces by the square root of the scalar curvature , Invent. Math. (1987),no.1, 63-82.[18] E. DiBenedetto and A. Friedman, H¨older estimates for nonlinear degenerate parabolic systems , J. reine angew.Math. (1985), 1-22.[19] E. Cabezas-Rivas and C. Sinestrari, Volume-preserving flow by powers of the mth mean curvature , Calc. Var.Partial Differential Equ., (2010), no. 3-4, 441–469.[20] C. Gerhardt, Curvature Problems , Series in Geometry and Topology, vol. , International Press, Somerville,MA, 2006.[21] C. Gerhardt, Curvature flows in the sphere , J. Differential Geom. (2015), no. 2, 301–347.[22] P. Guan and L. Ni, Entropy and a convergence theorem for Gauss curvature flow in high dimension , J. Eur.Math. Soc. (JEMS), (2017), no.12, 3735–3761.[23] G. Huisken, Flow by mean curvature of convex surfaces into spheres , J. Differential Geom. (1984), 237–266.[24] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, , Invent.Math. (1986), no.3, 463–480.[25] G. Huisken, Deforming hypersurfaces of the sphere by their mean curvature , Math. Z. (1987), no. 2,205–219[26] J.A. McCoy, Curvature contraction flows in the sphere , Proc. Amer. Math. Soc. (2018), 1243–1256.[27] O.C. Schn¨urer, Surfaces contracting with speed | A | , J. Differential Geom. (2005), no.3, 347–363.[28] F. Schulze, Convexity estimates for flows by powers of mean curvature, with an appendix by Felix Schulzeand Oliver C. Schn¨urer , Ann. Scuola Norm. Sup. Pisa Cl, Sci. (5) (2006), 261–277.[29] K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature , Comm. Pure Appl. Math. (1985),no.6, 867–882.[30] H. Yu, Dual flows in hyperbolic space and de Sitter space , arXiv:1604.02369v1. Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China E-mail address : [email protected] Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China E-mail address : [email protected] Mathematical Sciences Institute, Australian National University, Canberra, ACT 2601, Aus-tralia E-mail address : [email protected] Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China E-mail address ::