Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space
VVOLUME PRESERVING FLOW AND ALEXANDROV-FENCHEL TYPEINEQUALITIES IN HYPERBOLIC SPACE
BEN ANDREWS, XUZHONG CHEN, AND YONG WEI
Abstract.
In this paper, we study flows of hypersurfaces in hyperbolic space, and applythem to prove geometric inequalities. In the first part of the paper, we consider volumepreserving flows by a family of curvature functions including positive powers of k -th meancurvatures with k = 1 , · · · , n , and positive powers of p -th power sums S p with p >
0. Weprove that if the initial hypersurface M is smooth and closed and has positive sectionalcurvatures, then the solution M t of the flow has positive sectional curvature for any time t >
0, exists for all time and converges to a geodesic sphere exponentially in the smoothtopology. The convergence result can be used to show that certain Alexandrov-Fenchelquermassintegral inequalities, known previously for horospherically convex hypersurfaces,also hold under the weaker condition of positive sectional curvature.In the second part of this paper, we study curvature flows for strictly horosphericallyconvex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric,increasing and homogeneous degree one function f of the shifted principal curvatures λ i = κ i −
1, plus a global term chosen to impose a constraint on the quermassintegralsof the enclosed domain, where f is assumed to satisfy a certain condition on the secondderivatives. We prove that if the initial hypersurface is smooth, closed and strictlyhorospherically convex, then the solution of the flow exists for all time and converges to ageodesic sphere exponentially in the smooth topology. As applications of the convergenceresult, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces inhyperbolic space, and a new class of Alexandrov-Fenchel type inequalities for smoothhorospherically convex hypersurfaces in hyperbolic space. Contents
1. Introduction 22. Preliminaries 113. Preserving positive sectional curvature 154. Proof of Theorem 1.2 175. Horospherically convex regions 266. Proof of Theorem 1.7 32
Date : May 31, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Volume preserving flow, Alexandrov-Fenchel inequalities, Hyperbolic space,Horospherically convex hypersurfaces.This research was supported by Australian Laureate Fellowship FL150100126 of the Australian ResearchCouncil. The second author was also supported by the Fundamental Research Funds for the CentralUniversities. a r X i v : . [ m a t h . DG ] M a y B. ANDREWS, X. CHEN, AND Y. WEI
7. Conformal deformation in the conformal class of ¯ g Introduction
Let X : M n → H n +1 be a smooth embedding such that M = X ( M ) is a closed smoothhypersurface in the hyperbolic space H n +1 . We consider a smooth family of immersions X : M n × [0 , T ) → H n +1 satisfying ∂∂t X ( x, t ) = ( φ ( t ) − Ψ( x, t )) ν ( x, t ) ,X ( · ,
0) = X ( · ) , (1.1)where ν ( x, t ) is the unit outward normal of M t = X ( M, t ), Ψ is a smooth curvaturefunction evaluated at the point ( x, t ) of M t , the global term φ ( t ) is chosen to impose aconstraint on the enclosed volume or quermassintegrals of M t .The volume preserving mean curvature flow in hyperbolic space was first studied byCabezas-Rivas and Miquel [14] in 2007. By imposing horospherically convexity (the condi-tion that all principal curvatures exceed 1, which will also be called h-convex) on the initialhypersurface, they proved that the solution exists for all time and converges smoothly to ageodesic sphere. Some other mixed volume preserving flows were considered in [25,32] withspeed given by homogeneous degree one functions of the principal curvatures. RecentlyBertini and Pipoli [11] succeeded in treating flows by more general functions of mean curva-ture, including in particular any positive power of mean curvature. In a recent paper [10],the first and the third authors proved the smooth convergence of quermassintegral preserv-ing flows with speed given by any positive power of a homogeneous degree one function f ofthe principal curvatures for which the dual function f ∗ ( x , · · · , x n ) = ( f ( x − , · · · , x − n )) − is concave and approaches zero on the boundary of the positive cone. This includes inparticular the volume preserving flow by positive powers of k -th mean curvature for h-convex hypersurfaces. Note that in all the above mentioned work, the initial hypersurfaceis assumed to be h-convex.One reason to consider constrained flows of the kind considered here is to prove geomet-ric inequalities: In particular, the convergence of the volume-preserving mean curvatureflow to a sphere implies that the area of the initial hypersurface is no less than that ofa geodesic sphere with the same enclosed volume, since the area is non-increasing whilethe volume remains constant under the flow. The same motivation lies behind [32], whereinequalities between quermassintegrals were deduced from the convergence of certain flows.In this paper, we make the following contributions:(1) In the first part of the paper, we weaken the horospherical convexity condition,allowing instead hypersurfaces for which the intrinsic sectional curvatures are posi-tive. We consider the flow (1.1) for hypersurfaces with positive sectional curvatureand with speed Ψ given by any positive power of a smooth, symmetric, strictlyincreasing and homogeneous of degree one function of the Weingarten matrix W OLUME PRESERVING FLOW 3 of M t . Here we say a hypersurface M in hyperbolic space has positive sectionalcurvature if its sectional curvature R Mijij > ≤ i < j ≤ n , which byGauss equation is equivalent to the principal curvatures of M satisfying κ i κ j > ≤ i (cid:54) = j ≤ n . This is a weaker condition than h-convexity. As a consequencewe deduce inequalities between volume and other quermassintegrals for hypersur-faces with positive sectional curvature, extending inequalities previously knownonly for horospherically convex hypersurfaces.(2) In the second part of this paper, we consider flows (1.1) for strictly h-convex hy-persurfaces in which the speed Ψ is homogeneous as a function of the shifted Wein-garten matrix W −
I of M t , rather than the Weingarten matrix itself. Using theseflows we are able to prove a new class of integral inequalities for horosphericallyconvex hypersurfaces.(3) In order to understand these new functionals we introduce some new machineryfor horospherically convex regions, including a horospherical Gauss map and ahorospherical support function. We also develop an interesting connection (closelyrelated to the results of [15]) between flows of h-convex hypersurfaces in hyperbolicspace by functions of principal curvatures, and conformal flows of conformallyflat metrics on S n by functions of the eigenvalues of the Schouten tensor. Thisallows us to translate our results to convergence theorems for metric flows, andour isoperimetric inequalities to corresponding results for conformally flat metrics.We expect that these will prove useful in future work.We will describe our results in more detail in the rest of this section:1.1. Volume preserving flow with positive sectional curvature.
Suppose that theinitial hypersurface M has positive sectional curvature. We consider the smooth familyof immersions X : M n × [0 , T ) → H n +1 satisfying ∂∂t X ( x, t ) = ( φ ( t ) − F α ( W )) ν ( x, t ) ,X ( · ,
0) = X ( · ) , (1.2)where α > ν ( x, t ) is the unit outward normal of M t = X ( M, t ), F is a smooth, symmet-ric, strictly increasing and homogeneous of degree one function of the Weingarten matrix W of M t . The global term φ ( t ) in (1.2) is defined by φ ( t ) = 1 | M t | (cid:90) M t F α dµ t (1.3)such that the volume of Ω t remains constant along the flow (1.2), where dµ t is the areameasure on M t with respect to the induced metric.Since F ( W ) is symmetric with respect to the components of W , by a theorem of Schwarz[28] we can write F ( W ) = f ( κ ) as a symmetric function of the eigenvalues of W . Weassume that f satisfies the following assumption: Assumption 1.1.
Suppose f is a smooth symmetric function defined on the positive cone Γ + := { κ = ( κ , · · · , κ n ) ∈ R n : κ i > , ∀ i = 1 , · · · , n } , and satisfies (i) f is positive, strictly increasing, homogeneous of degree one and is normalized suchthat f (1 , · · · ,
1) = 1 ; B. ANDREWS, X. CHEN, AND Y. WEI (ii)
For any i (cid:54) = j , ( ∂f∂κ i κ i − ∂f∂κ j κ j )( κ i − κ j ) ≥ . (1.4)(iii) For all ( y , · · · , y n ) ∈ R n , (cid:88) i,j ∂ log f∂κ i ∂κ j y i y j + n (cid:88) i =1 κ i ∂ log f∂κ i y i ≥ . (1.5)Examples satisfying Assumption 1.1 include f = n − /k S /kk ( k >
0) and f = E /kk (see,e.g., [16, 17]), where E k = (cid:18) nk (cid:19) − σ k ( κ ) = (cid:18) nk (cid:19) − (cid:88) ≤ i < ···
0. The inequalities (1.4) and (1.5) are equivalent to the statement thatlog F is a convex function of the components of log W , which is the map with the sameeigenvectors as W and eigenvalues log κ i . In particular, if f and f are two symmetricfunctions satisfying (1.4) and (1.5), then the function f α with α > f f also satisfy (1.4) and (1.5). Note that the Cauchy-Schwarz inequality and (1.5) imply thatany symmetric function f satisfying (1.5) must be inverse concave, i.e., its dual function f ∗ ( z , · · · , z n ) = f ( z − , · · · , z − n ) − is concave with respect to its argument.The first result of this paper is the following convergence result for the flow (1.2): Theorem 1.2.
Let X : M n → H n +1 be a smooth embedding such that M = X ( M ) isa closed hypersurface in H n +1 ( n ≥ with positive sectional curvature. Assume that f satisfies Assumption 1.1, and either (i) f ∗ vanishes on the boundary of Γ + , and lim x → f ( x, x , · · · , x ) = + ∞ , (1.6) and α > , or (ii) n = 2 , f = ( κ κ ) / and α ∈ [1 / , .Then the flow (1.2) with global term φ ( t ) given by (1.3) has a smooth solution M t forall time t ∈ [0 , ∞ ) , and M t has positive sectional curvature for each t > and convergessmoothly and exponentially to a geodesic sphere of radius r ∞ determined by Vol( B ( r ∞ )) =Vol(Ω ) as t → ∞ .Remark . Examples of function f satisfying Assumption 1.1 and the condition (i) ofTheorem 1.2 include:a). n ≥ f = n − /k S /kk with k > n ≥ f = E /kk with k = 1 , · · · , n ;c). n = 2 , f = ( κ + κ ) / OLUME PRESERVING FLOW 5
Remark . We remark that the contracting curvature flows for surfaces with positivescalar curvature in hyperbolic 3-space H have been studied by the first two authors in arecent work [7].As a key step in the proof of Theorem 1.2, we prove in § M t is preserved along the flow (1.2) with any f satisfying Assumption 1.1 and any α >
0. In order to show that the positivity ofsectional curvatures are preserved, we consider the sectional curvature as a function onthe frame bundle O ( M ) over M , and apply a maximum principle. This requires a ratherdelicate computation, using inequalities on the Hessian on the total space of O ( M ) toshow the required inequality on the time derivative at a minimum point. The argument isrelated to that used by the first author to prove a generalised tensor maximum principlein [4, Theorem 3.2], but cannot be deduced directly from that result. The argumentcombines the ideas of the generalised tensor maximum principle with those of vectorbundle maximum principles for reaction-diffusion equations [8, 20].We remark that the flow (1.2) with f = (cid:18) E k E l (cid:19) k − l , ≤ l < k ≤ n (1.7)and any power α > §
4. In § t along the flow (1.2). Recall that the inner radius ρ − and outer radius ρ + of a boundeddomain Ω are defined as ρ − = sup (cid:91) p ∈ Ω { ρ > B ρ ( p ) ⊂ Ω } ρ + = inf (cid:91) p ∈ Ω { ρ > ⊂ B ρ ( p ) } , where B ρ ( p ) denotes the geodesic ball of radius ρ and centered at some point p in thehyperbolic space. All the previous papers [10, 11, 14, 25, 32] on constrained curvature flowsin hyperbolic space focus on horospherically convex domains, which have the propertythat ρ + ≤ c ( ρ − + ρ / − ), see e.g. [14, 25]. However, no such property is known for hyper-surfaces with positive sectional curvature. Our idea to overcome this obstacle is to use anAlexandrov reflection argument to bound the diameter of the domain Ω t enclosed by theflow hypersurface M t . Then we project the domain Ω t to the unit ball in Euclidean space R n +1 via the Klein model of the hyperbolic space. The upper bound on the diameter ofΩ t implies that this map has bounded distortion. This together with the preservation ofthe volume of Ω t gives a uniform lower bound on the inner radius of Ω t .Then in § f satisfies Assumption 1.1, where the positivity of sectional curvatures of M t will beused to estimate the zero order terms of the evolution equation of the auxiliary function.In § f together withthe positivity of sectional curvatures imply the uniform two-sides positive bound of theprincipal curvatures of M t . In the case (ii) of Theorem 1.2, the estimate 1 ≤ κ κ = B. ANDREWS, X. CHEN, AND Y. WEI f ( κ ) ≤ C does not prevent κ from going to infinity. Instead, we will obtain the estimateon the pinching ratio κ /κ by applying the maximum principle to the evolution equationof G ( κ , κ ) = ( κ κ ) α − ( κ − κ ) with α ∈ [1 / , R . Once we have the uniform estimate on the principal curvaturesof the evolving hypersurfaces, higher regularity estimates can be derived by a standardargument. A continuation argument then yields the long time existence of the flow andthe Alexandrov reflection argument as in [10, §
6] implies the smooth convergence of theflow to a geodesic sphere.1.2.
Alexandrov-Fenchel inequalities.
The volume preserving curvature flow is a use-ful tool in the study of hypersurface geometry. We will illustrate an application of Theorem1.2 in the proof of Alexandrov-Fenchel type inequalities (involving the quermassintegrals)for hypersurfaces in hyperbolic space. Recall that for a convex domain Ω in hyperbolicspace, the quermassintegral W k (Ω) is defined as follows (see [26, 27]): W k (Ω) = ω k − · · · ω ω n − · · · ω n − k (cid:90) L k χ ( L k ∩ Ω) dL k , k = 1 , · · · , n, (1.8)where L k is the space of k -dimensional totally geodesic subspaces L k in H n +1 and ω n denotes the area of n -dimensional unit sphere in Euclidean space. The function χ isdefined to be 1 if L k ∩ Ω (cid:54) = ∅ and to be 0 otherwise. Furthermore, we set W (Ω) = | Ω | , W n +1 (Ω) = | B n +1 | = ω n n + 1 . If the boundary of Ω is smooth, we can define the principal curvatures κ = ( κ , · · · , κ n )and the curvature integrals V n − k (Ω) = (cid:90) ∂ Ω E k ( κ ) dµ, k = 0 , , · · · , n (1.9)of the boundary M = ∂ Ω. The quermassintegrals and the curvature integrals of a smoothconvex domain Ω in H n +1 are related by the following equations (see [27]): V n − k (Ω) = ( n − k ) W k +1 (Ω) + kW k − (Ω) , k = 1 , · · · , n (1.10) V n (Ω) = nW (Ω) = | ∂ Ω | . (1.11)In [32], Wang and Xia proved the Alexandrov-Fenchel inequalities for smooth h-convexdomain Ω in H n +1 , which states that there holds W k (Ω) ≥ f k ◦ f − l ( W l (Ω)) (1.12)for any 0 ≤ l < k ≤ n , with equality if and only if Ω is a geodesic ball, where f k : R + → R + is an increasing function defined by f k ( r ) = W k ( B ( r )), the k -th quermassintegral of thegeodesic ball of radius r . The proof in [32] is by applying the quermassintegral preservingflow for smooth h-convex hypersurfaces with speed given by the quotient (1.7) and α = 1, Note that the definition (1.8) is different with the definition given in [27] by a constant multiple n +1 − kn +1 . OLUME PRESERVING FLOW 7 and is similar with the Euclidean analogue considered by McCoy [23]. The inequality(1.12) can imply the following inequality (cid:90) ∂ Ω E k dµ ≥ | ∂ Ω | (cid:32) (cid:18) | ∂ Ω | ω n (cid:19) − /n (cid:33) k/ (1.13)for smooth h-convex domains, which compares the curvature integral (1.9) and the bound-ary area. Note that the inequality (1.13) with k = 2 was proved earlier by the third authorwith Li and Xiong [22] for star-shaped and 2-convex domains using the inverse curvatureflow in hyperbolic space. For the other even k , the inequality (1.13) was also proved forsmooth h-convex domains using the inverse curvature flow by Ge, Wang and Wu [18]. It’san interesting problem to prove the inequalities (1.12) and (1.13) under an assumptionthat is weaker than h-convexity.Applying the result in Theorem 1.2, we show that the h-convexity assumption for theinequality (1.12) can be replaced by the weaker assumption of positive sectional curvature in the case l = 0 and 1 ≤ k ≤ n . Corollary 1.5.
Let M = ∂ Ω be a smooth closed hypersurface in H n +1 which has positivesectional curvature and encloses a smooth bounded domain Ω . Then for any n ≥ and k = 1 , · · · , n , we have W k (Ω) ≥ f k ◦ f − ( W (Ω)) , (1.14) where f k : R + → R + is an increasing function defined by f k ( r ) = W k ( B ( r )) , the k -thquermassintegral of the geodesic ball of radius r . Moreover, equality holds in (1.14) if andonly if Ω is a geodesic ball. The quermassintegral W k (Ω t ) of the evolving domain Ω t along the flow (1.2) with F = E /kk satisfies (see Lemma 2.3) ddt W k (Ω t ) = (cid:90) M t E k ( φ ( t ) − E α/kk ) dµ t , which is non-positive for each α > φ ( t ) and the H¨older inequality.This means that W k (Ω t ) is monotone decreasing along the flow (1.2) with F = E /kk unless E k is constant on M t (which is equivalent to M t being a geodesic sphere). Then Corollary1.5 follows from the monotonicity of W k and the convergence result in Theorem 1.2.1.3. Volume preserving flow for horospherically convex hypersurfaces.
In thesecond part of this paper, we will consider the flow of h-convex hypersurfaces in hyperbolicspace with speed given by functions of the shifted Weingarten matrix
W −
I plus a globalterm chosen to preserve modified quermassintegrals of the evolving domains. Let us firstdefine the following modified quermassintegrals: (cid:102) W k (Ω) := k (cid:88) i =0 ( − k − i (cid:18) ki (cid:19) W i (Ω) , k = 0 , · · · , n, (1.15)for a h-convex domain Ω in hyperbolic space. Thus (cid:102) W k is a linear combination of thequermassintegrals of Ω. In particular, (cid:102) W (Ω) = | Ω | is the volume of Ω. The modifiedquermassintegrals defined in (1.15) satisfy the following property: B. ANDREWS, X. CHEN, AND Y. WEI
Proposition 1.6.
The modified quermassintegral (cid:102) W k is monotone with respect to inclusionfor h-convex domains: That is, if Ω and Ω are h-convex domains with Ω ⊂ Ω , then (cid:102) W k (Ω ) ≤ (cid:102) W k (Ω ) . This property is not obvious from the definition (1.15) and its proof will be given in §
5. We will first investigate some of the properties of horospherically convex regions inhyperbolic space H n +1 . In particular, for such regions we define a horospherical Gaussmap , which is a map to the unit sphere, and we show that each horospherically convexregion is completely described in terms of a scalar function u on the sphere S n which wecall the horospherical support function . There are interesting formal similarities betweenthis situation and that of convex Euclidean bodies. We show that the h-convexity of aregion Ω is equivalent to that the following matrix A ij = ¯ ∇ j ¯ ∇ k ϕ − | ¯ ∇ ϕ | ϕ ¯ g ij + ϕ − ϕ − g ij on the sphere S n is positive definite, where ¯ g ij is the standard round metric on S n , ϕ = e u and u is the horospherical support function of Ω. The shifted Weingarten matrix W −
Iis related to the matrix A ij by A ij = ϕ − (cid:104) ( W − I) − (cid:105) ki ¯ g kj . (1.16)Using this characterization of h-convex domains, for any two h-convex domains Ω andΩ with Ω ⊂ Ω we can find a foliation of h-convex domains Ω t which is expanding fromΩ to Ω . This can be used to prove Proposition 1.6 by computing the variation of (cid:102) W k .We expect that the description of horospherically convex regions which we develop herewill be useful in further investigations beyond the scope of this paper.The flow we will consider is the following: ∂∂t X ( x, t ) = ( φ ( t ) − F ( W −
I)) ν ( x, t ) ,X ( · ,
0) = X ( · ) (1.17)for smooth and strictly h-convex hypersurface in hyperbolic space, where F is a smooth,symmetric, homegeneous of degree one function of the shifted Weingarten matrix W −
I =( h ji − δ ji ). For simplicity, we denote S ij = h ji − δ ji . Note that the eigenvalues of ( S ij )are the shifted principal curvatures λ = ( λ , · · · , λ n ) = ( κ − , · · · , κ n − F ( W −
I) = f ( λ ), where f is a smooth symmetric function of n variables λ = ( λ , · · · , λ n ). We choose the global term φ ( t ) in (1.17) as φ ( t ) = (cid:18)(cid:90) M t E l ( λ ) dµ t (cid:19) − (cid:90) M t E l ( λ ) F dµ t , l = 0 , · · · , n (1.18)such that (cid:102) W l (Ω t ) remains constant, where Ω t is the domain enclosed by the evolvinghypersurface M t .We will prove the following result for the flow (1.17) with φ ( t ) given in (1.18). Theorem 1.7.
Let n ≥ and X : M n → H n +1 be a smooth embedding such that M = X ( M ) is a smooth closed and strictly h-convex hypersurface in H n +1 . If f is asmooth, symmetric, increasing and homogeneous of degree one function, and either OLUME PRESERVING FLOW 9 (i) f is concave and f approaches zero on the boundary of the positive cone Γ + , or (ii) f is concave and inverse concave, or (iii) f is inverse concave and its dual function f ∗ approaches zero on the boundary ofpositive cone Γ + , or (iv) n = 2 ,then the flow (1.17) with the global term φ ( t ) given by (1.18) has a smooth solution M t for all time t ∈ [0 , ∞ ) , and M t is strictly h-convex for any t > and converges smoothlyand exponentially to a geodesic sphere of radius r ∞ determined by (cid:102) W l ( B ( r ∞ )) = (cid:102) W l (Ω ) as t → ∞ . Constrained curvature flows in hyperbolic space by homogeneous of degree one, concaveand inverse concave function of the principal curvatures were studied by Makowski [25]and Wang and Xia in [32]. The quermassintegral preserving flow by any positive power ofa homogeneous of degree one function of the principal curvatures, which is inverse concaveand its dual function f ∗ approaches zero on the boundary of positive cone Γ + , was studiedrecently by the first and the third authors in [10]. Note that the speed function f of theflow (1.17) in Theorem 1.7 is not a homogeneous function of the principal curvatures κ i and there are essential differences in the analysis compared with the previously mentionedwork [10, 25, 32].The key step in the proof of Theorem 1.7 is a pinching estimate for the shifted principalcurvatures λ i . That is, we will show that the ratio of the largest shifted principal curvature λ n to the smallest shifted principal curvature λ is controlled by its initial value along theflow (1.17). For the proof, we adapt methods from the proof of pinching estimates ofthe principal curvatures for contracting curvature flows [1, 4, 5, 9] and the constrainedcurvature flows in Euclidean space [23, 24]. In particular, in the case (iii) we define thetensor T ij = S ij − εF δ ji and show that the positivity of T ij is preserved by applying thetensor maximum principle (proved by the first author in [4]). The inverse concavity isused to estimate the sign of the gradient terms. This case is similar with the pinchingestimate for the contracting curvature flow in Euclidean case in [9, Lemma 11]. Althoughthe proof there is given in terms of the Gauss map parametrisation of the convex solutionsof the flow in Euclidean space, which is not available in hyperbolic space, we can deal withthe gradient terms directly using the inverse concavity property of f .To prove Theorem 1.7, we next show that the inner radius and outer radius of theenclosed domain Ω t of the evolving hypersurface M t satisfies a uniform estimate 0 Let M be a smooth, closed and strictly h-convex hypersurface in H n +1 with principal curvatures κ = ( κ , · · · , κ n ) satisfying f ( λ ) = C for some constant C > ,where λ = ( λ , · · · , λ i ) with λ i = κ i − and f is a symmetric function satisfying thecondition in Theorem 1.7. Then M is a geodesic sphere. The second application of Theorem 1.7 is a new class of Alexandrov-Fenchel type in-equalities between quermassintegrals of h-convex hypersurface in hyperbolic space. Corollary 1.9. Let M = ∂ Ω be a smooth, closed and strictly h-convex hypersurface in H n +1 . Then for any ≤ l < k ≤ n , there holds (cid:102) W k (Ω) ≥ ˜ f k ◦ ˜ f − l ( (cid:102) W l (Ω)) , (1.19) with equality holding if and only if Ω is a geodesic ball. Here the function ˜ f k : R + → R + is defined by ˜ f k ( r ) = (cid:102) W k ( B ( r )) , which is an increasing function by Proposition 1.6. ˜ f − l is the inverse function of ˜ f l . The inequality (1.19) can be obtained by applying Theorem 1.7 with f chosen as f = (cid:18) E k ( λ ) E l ( λ ) (cid:19) k − l , ≤ l < k ≤ n (1.20)in the flow (1.17). We see that along the flow (1.17) with such f , the modified quermass-integral (cid:102) W l (Ω t ) remains a constant and (cid:102) W k (Ω t ) is monotone decreasing in time by theH¨older inequality. In fact, by Lemma 2.4 the modified quermassintegral evolves by ddt (cid:102) W k (Ω t ) = (cid:90) M t E k ( λ ) (cid:32) φ ( t ) − (cid:18) E k ( λ ) E l ( λ ) (cid:19) k − l (cid:33) dµ t . (1.21)Applying the H¨older inequality to the equation (1.21) yields that (cid:102) W k (Ω t ) is monotonedecreasing in time unless E k ( λ ) = CE l ( λ ) on M t (which is equivalent to that M t is ageodesic sphere by Corollary 1.8). Since the flow exists for all time and converges to ageodesic sphere B r , the inequality (1.19) follows from the monotonicity of (cid:102) W k (Ω t ) and thepreservation of (cid:102) W l (Ω t ). Remark . We remark that the inequalities (1.19) are new and can be viewed as animprovement of the inequalities (1.12). For example, the inequality (1.19) with l = 0implies that k (cid:88) i =0 ( − k − i (cid:18) ki (cid:19)(cid:18) W i (Ω) − f i ◦ f − ( W (Ω)) (cid:19) ≥ . (1.22)By induction on k , (1.22) implies that each W i (Ω) − f i ◦ f − ( W (Ω)) is nonnegative for h -convex domains. Thus our inequalities (1.19) imply that the linear combinations of W i (Ω) − f i ◦ f − ( W (Ω)) as in (1.22) are also nonnegative for h-convex domains. Acknowledgments. The second author is grateful to the Mathematical Sciences Instituteat the Australian National University for its hospitality during his visit, when some of thiswork was completed. OLUME PRESERVING FLOW 11 Preliminaries In this section we collect some properties of smooth symmetric functions f of n variables,and recall the evolution equations of geometric quantities along the flows (1.2) and (1.17).2.1. Properties of symmetric functions. For a smooth symmetric function F ( A ) = f ( κ ( A )), where A = ( A ij ) ∈ Sym( n ) is symmetric matrix and κ ( A ) = ( κ , · · · , κ n ) givethe eigenvalues of A , we denote by ˙ F ij and ¨ F ij,kl the first and second derivatives of F with respect to the components of its argument, so that ∂∂s F ( A + sB ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ˙ F ij ( A ) B ij and ∂ ∂s F ( A + sB ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ¨ F ij,kl ( A ) B ij B kl for any two symmetric matrixs A, B . We also use the notation˙ f i ( κ ) = ∂f∂κ i ( κ ) , ¨ f ij ( κ ) = ∂ f∂κ i ∂κ j ( κ ) . for the first and the second derivatives of f with respect to κ . At any diagonal A withdistinct eigenvalues κ = κ ( A ), the first derivatives of F satisfy˙ F ij ( A ) = ˙ f i ( κ ) δ ij and the second derivative of F in direction B ∈ Sym( n ) is given in terms of ˙ f and ¨ f by(see [4]): ¨ F ij,kl ( A ) B ij B kl = (cid:88) i,j ¨ f ij ( κ ) B ii B jj + 2 (cid:88) i>j ˙ f i ( κ ) − ˙ f j ( κ ) κ i − κ j B ij . (2.1)This formula makes sense as a limit in the case of any repeated values of κ i .From the equation (2.1), we have Lemma 2.1. Suppose A has distinct eigenvalues κ = κ ( A ) . Then F is concave at A ifand only if f is concave at κ and (cid:16) ˙ f k − ˙ f l (cid:17) ( κ k − κ l ) ≤ , ∀ k (cid:54) = l. (2.2)In this paper, we also need the inverse concavity property of f in many cases. Weinclude the properties of inverse concave function in the following lemma. Lemma 2.2 ([4, 10]) . (i) If f is inverse concave, then n (cid:88) k,l =1 ¨ f kl y k y l + 2 n (cid:88) k =1 ˙ f k κ k y k ≥ f − ( n (cid:88) k =1 ˙ f k y k ) (2.3) for any y = ( y , · · · , y n ) ∈ R n , and ˙ f k − ˙ f l κ k − κ l + ˙ f k κ l + ˙ f l κ k ≥ , ∀ k (cid:54) = l. (2.4) (ii) If f = f ( κ , · · · , κ n ) is inverse concave, then n (cid:88) i =1 ˙ f i κ i ≥ f . (2.5)2.2. Evolution equations. Along any smooth flow ∂∂t X ( x, t ) = ϕ ( x, t ) ν ( x, t ) (2.6)of hypersurfaces in hyperbolic space H n +1 , where ϕ is a smooth function on the evolvinghypersurfaces M t = X ( M n , t ), we have the following evolution equations on the inducedmetric g ij , the induced area element dµ t and the Weingarten matrix W = ( h ji ) of M t : ∂∂t g ij = 2 ϕh ij (2.7) ∂∂t dµ t = nE ϕdµ t (2.8) ∂∂t h ji = − ∇ j ∇ i ϕ − ϕ ( h ki h jk − δ ji ) (2.9)From the evolution equations (2.8) and (2.9), we can derive the evolution equation of thecurvature integral V n − k : ddt V n − k (Ω t ) = ddt (cid:90) M t E k dµ t = (cid:90) M t (cid:18) ∂∂t E k + nE E k ϕ (cid:19) dµ t = (cid:90) M t (cid:32) − ∂E k ∂h ji ∇ j ∇ i ϕ − ϕ ∂E k ∂h ji ( h ki h jk − δ ji ) + nE E k ϕ (cid:33) dµ t = (cid:90) M t ϕ (cid:18) ( n − k ) E k +1 + kE k − (cid:19) dµ t , (2.10)where we used integration by part and the fact that ∂E k ∂h ji is divergence free. Since thequermassintegrals are related to the curvature integrals by the equations (1.10) and (1.11),applying an induction argument to the equation (2.10) yields that Lemma 2.3 (cf.[10, 32]) . Along the flow (2.6) , the quermassintegral W k of the evolvingdomain Ω t satisfies ddt W k (Ω t ) = (cid:90) M t E k ( κ ) ϕdµ t , k = 0 , · · · , n. We can also derive the following evolution equation for the modified quermassintegrals. Lemma 2.4. Along the flow (2.6) , the modified quermassintegral (cid:102) W k of the evolving do-main Ω t satisfies ddt (cid:102) W k (Ω t ) = (cid:90) M t E k ( λ ) ϕdµ t , k = 0 , · · · , n, where λ = ( λ , · · · , λ n ) = ( κ − , · · · , κ n − are the shifted principal curvatures of M t . OLUME PRESERVING FLOW 13 Proof. Firstly, we derive the formula for σ k ( λ ) in terms of σ i ( κ ) , i = 0 , · · · , k . By thedefinition of the elementary symmetric polynomial, we have n (cid:89) i =1 ( t + λ i ) = n (cid:88) k =0 σ k ( λ ) t n − k . On the other hand, n (cid:89) i =1 ( t + λ i ) = n (cid:89) i =1 ( t − κ i ) = n (cid:88) l =0 σ l ( κ )( t − n − l = n (cid:88) l =0 σ l ( κ ) n − l (cid:88) i =0 (cid:18) n − li (cid:19) t i ( − n − l − i = n (cid:88) k =0 (cid:32) k (cid:88) i =0 (cid:18) n − ik − i (cid:19) ( − k − i σ i ( κ ) (cid:33) t n − k . Comparing the coefficients of t n − k , we have σ k ( λ ) = k (cid:88) i =0 (cid:18) n − ik − i (cid:19) ( − k − i σ i ( κ )= k (cid:88) i =0 (cid:18) n − ik − i (cid:19) ( − k − i (cid:18) ni (cid:19) E i ( κ )= (cid:18) nk (cid:19) k (cid:88) i =0 ( − k − i (cid:18) ki (cid:19) E i ( κ ) . Equivalently, we have E k ( λ ) = (cid:18) nk (cid:19) − σ k ( λ ) = k (cid:88) i =0 ( − k − i (cid:18) ki (cid:19) E i ( κ ) . (2.11)Then by the definition (1.15) of (cid:102) W k and Lemma 2.3, ddt (cid:102) W k (Ω t ) = k (cid:88) i =0 ( − k − i (cid:18) ki (cid:19) ddt W i (Ω t )= k (cid:88) i =0 ( − k − i (cid:18) ki (cid:19) (cid:90) M t E i ( κ ) ϕdµ t = (cid:90) M t E k ( λ ) ϕdµ t . (cid:3) If we consider the flow (1.2), i.e., ϕ = φ ( t ) − Ψ( W ), using (2.9) and the Simons’ identitywe have the evolution equations for the curvature function Ψ = Ψ( W ) and the Weingartenmatrix W = ( h ji ) of M t (see [10]): ∂∂t Ψ = ˙Ψ kl ∇ k ∇ l Ψ + (Ψ − φ ( t ))( ˙Ψ ij h ki h jk − ˙Ψ ij δ ji ) , (2.12) and ∂∂t h ji = ˙Ψ kl ∇ k ∇ l h ji + ¨Ψ kl,pq ∇ i h kl ∇ j h pq + ( ˙Ψ kl h rk h rl + ˙Ψ kl g kl ) h ji − ˙Ψ kl h kl ( h pi h jp + δ ji ) + (Ψ − φ ( t ))( h ki h jk − δ ji ) , (2.13)where ∇ denotes the Levi-Civita connection with respect to the induced metric g ij on M t , and ˙Ψ kl , ¨Ψ kl,pq denote the derivatives of Ψ with respect to the components of theWeingarten matrix W = ( h ji ).If we consider the flow (1.17) of h-convex hypersurfaces, i.e., ϕ = φ ( t ) − F ( W − I), wehave the similar evolution equation for the curvature function F∂∂t F = ˙ F kl ∇ k ∇ l F + ( F − φ ( t ))( ˙ F ij h ki h jk − ˙ F ij δ ji ) , (2.14)and a parabolic type equation for the Weingarten matrix W = ( h ji ) of M t : ∂∂t h ji = ˙ F kl ∇ k ∇ l h ji + ¨ F kl,pq ∇ i h kl ∇ j h pq + ( ˙ F kl h rk h rl + ˙ F kl g kl ) h ji − ˙ F kl h kl ( h pi h jp + δ ji ) + ( F − φ ( t ))( h ki h jk − δ ji ) . (2.15)However, we observe that ˙ F kl , ¨ F kl,pq in (2.14) and (2.15) denote the derivatives of F withrespect to the components of shifted Weingarten matrix W − I, so the homogeneity of F implies that ˙ F kl ( h lk − δ lk ) = F . Denote the components of the shifted Weingarten matrixby S ij = h ji − δ ji . Then the equation (2.15) implies that ∂∂t S ij = ˙ F kl ∇ k ∇ l S ij + ¨ F kl,pq ∇ i h kl ∇ j h pq + ( ˙ F kl S kr S rl + 2 F − φ ( t )) S ij − ( φ ( t ) + ˙ F kl δ lk ) S ik S kj + ˙ F kl S kr S rl δ ji . (2.16)2.3. A generalised maximum principle. In § Theorem 2.5 ([4]) . Let S ij be a smooth time-varying symmetric tensor field on a compactmanifold M , satisfying ∂∂t S ij = a kl ∇ k ∇ l S ij + u k ∇ k S ij + N ij , where a kl and u are smooth, ∇ is a (possibly time-dependent) smooth symmetric connec-tion, and a kl is positive definite everywhere. Suppose that N ij v i v j + sup Λ a kl (cid:0) pk ∇ l S ip v i − Λ pk Λ ql S pq (cid:1) ≥ whenever S ij ≥ and S ij v j = 0 . If S ij is positive definite everywhere on M at t = 0 andon ∂M for ≤ t ≤ T , then it is positive on M × [0 , T ] . OLUME PRESERVING FLOW 15 Preserving positive sectional curvature In this section, we will prove that the flow (1.2) preserves the positivity of sectionalcurvatures, if α > f satisfies Assumption 1.1. Theorem 3.1. If the initial hypersurface M has positive sectional curvature, then alongthe flow (1.2) in H n +1 with f satisfying Assumption 1.1 and any power α > the evolvinghypersurface M t has positive sectional curvature for t > .Proof. The sectional curvature defines a smooth function on the Grassmannian bundleof two-dimensional subspaces of T M . For convenience we lift this to a function on theorthonormal frame bundle O ( M ) over M : Given a point x ∈ M and t ≥ 0, and a frame O = { e , · · · , e n } for T x M which is orthonormal with respect to g ( x, t ), we define G ( x, t, O ) = h ( x,t ) ( e , e ) h ( x,t ) ( e , e ) − h ( x,t ) ( e , e ) − . We consider a point ( x , t ) and a frame O = { ¯ e , · · · , ¯ e n } at which a new minimum ofthe function G is attained, so that we have G ( x, t, O ) ≥ G ( x , t , O ) for all x ∈ M andall t ∈ [0 , t ], and all O ∈ F ( M ) ( x,t ) . The fact that O achieves the minimum of G overthe fibre F ( M ) ( x ,t ) implies that e and e are eigenvectors of h ( x .t ) corresponding to κ and κ , where κ ≤ κ ≤ · · · ≤ κ n are the principal curvatures at ( x , t ). Since G is invariant under rotation in the subspace orthogonal to ¯ e and ¯ e , we can assume that h (¯ e i , ¯ e i ) = κ i and h (¯ e i , ¯ e j ) = 0 for i (cid:54) = j .The time derivative of G at ( x , t , O ) is given by Equation (2.13), noting that theframe O ( t ) for T x M defined by ddt e i ( t ) = ( F α − φ ) W ( e i ) remains orthonormal with respectto g ( x, t ) if e i ( t ) = ¯ e i for each i . This yields the following: ∂∂t G | ( x ,t , O ) = κ ∂∂t h + κ ∂∂ t h = κ ˙Ψ kl ∇ k ∇ l h + κ ˙Ψ kl ∇ k ∇ l h + κ ¨Ψ( ∇ h, ∇ h ) + κ ¨Ψ( ∇ h, ∇ h )+ 2 (cid:16) ˙Ψ kl h rk h rl + ˙Ψ kl g kl (cid:17) κ κ − ( α − κ κ ( κ + κ ) − ( α + 1)Ψ( κ + κ ) − φ ( t )( κ κ − κ + κ ) . (3.1)Since Ψ = f α , we have the following:2 (cid:16) ˙Ψ kl h rk h rl + ˙Ψ kl g kl (cid:17) κ κ − ( α − κ κ ( κ + κ ) − ( α + 1)Ψ( κ + κ ) − φ ( t )( κ κ − κ + κ )= 2 αf α − (cid:88) k ˙ f k ( κ k − κ )( κ k − κ ) + G (cid:32) f α − (cid:88) k ˙ f k κ k (2 ακ k − ( α − κ + κ )) − φ ( t )( κ + κ ) (cid:33) ≥ − CG, where C is a bound for the smooth function in the last bracket. To estimate the remainingterms, we consider the second derivatives of G along a curve on O ( M ) defined as follows:We let γ be any geodesic of g ( t ) in M with γ (0) = x , and define a frame O ( s ) =( e ( s ) , · · · , e n ( s )) at γ ( s ) by taking e i (0) = ¯ e i for each i , and ∇ s e i ( s ) = Γ ij e j ( s ) for some constant antisymmetric matrix Γ. Then we compute d ds G ( x ( s ) , t , O ( s )) (cid:12)(cid:12)(cid:12) s =0 = κ ∇ s h + κ ∇ s h + 2 (cid:0) ∇ s h ∇ s h − ( ∇ s h ) (cid:1) + 4 (cid:88) p> Γ p κ ∇ s h p + 4 (cid:88) p> Γ p κ ∇ s h p + 2 (cid:88) p> Γ p κ ( κ p − κ ) + 2 (cid:88) p> Γ p κ ( κ p − κ ) . (3.2)Since G has a minimum at ( x , t , O ), the right-hand side of (3.2) is non-negative for anychoice of Γ. Minimizing over Γ gives0 ≤ κ ∇ s h + κ ∇ s h + 2 (cid:0) ∇ s h ∇ s h − ( ∇ s h ) (cid:1) − (cid:88) p> κ κ p − κ ( ∇ s h p ) − (cid:88) p> κ κ p − κ ( ∇ s h p ) (3.3)where we terms on the last line as vanishing if the denominators vanish (since the corre-sponding component of ∇ h vanishes in that case). This gives ∂∂t G | ( x ,t , O ) ≥ κ ¨Ψ( ∇ h, ∇ h ) + κ ¨Ψ( ∇ h, ∇ h ) − (cid:88) k ˙Ψ k (cid:0) ∇ k h ∇ k h − ( ∇ k h ) (cid:1) + 2 (cid:88) k ˙Ψ k (cid:88) p> κ κ p − κ ( ∇ k h p ) + (cid:88) p> κ κ p − κ ( ∇ k h p ) − CG (3.4)The right-hand side can be expanded using Ψ = f α and the identity (2.1): f − α α (cid:18) ddt G + CG (cid:19) ≥ κ (cid:88) k,l ¨ f kl ∇ h kk ∇ h ll + ( α − 1) ( ∇ f ) f + (cid:88) k (cid:54) = l ˙ f k − ˙ f l κ k − κ l ( ∇ h kl ) + κ (cid:88) k,l ¨ f kl ∇ h kk ∇ h ll + ( α − 1) ( ∇ f ) f + (cid:88) k (cid:54) = l ˙ f k − ˙ f l κ k − κ l ( ∇ h kl ) − (cid:88) k ˙ f k (cid:0) ∇ k h ∇ k h − ( ∇ k h ) (cid:1) + 2 (cid:88) k ˙ f k (cid:88) p> κ κ p − κ ( ∇ k h p ) + (cid:88) p> κ κ p − κ ( ∇ k h p ) . Note that by assumption the function f satisfies the inequalities (1.4) and (1.5). By theinequality (1.4), for any k (cid:54) = l we have˙ f k − ˙ f l κ k − κ l + ˙ f k κ l = ˙ f k κ k − ˙ f l κ l ( κ k − κ l ) κ l ≥ (cid:88) k,l ¨ f kl y k y l ≥ f − ( n (cid:88) k =1 ˙ f k y k ) − n (cid:88) k =1 ˙ f k κ k y k OLUME PRESERVING FLOW 17 for all ( y , · · · , y n ) ∈ R n . These imply that f − α α (cid:18) dGdt + CG (cid:19) ≥ κ α ( ∇ f ) f − n (cid:88) k =1 ˙ f k κ k ( ∇ h kk ) − (cid:88) k (cid:54) = l ˙ f k κ l ( ∇ h kl ) + κ α ( ∇ f ) f − n (cid:88) k =1 ˙ f k κ k ( ∇ h kk ) − (cid:88) k (cid:54) = l ˙ f k κ l ( ∇ h kl ) + 2 (cid:88) k ˙ f k (cid:0) −∇ k h ∇ k h + ( ∇ k h ) (cid:1) + 2 n (cid:88) k =1 (cid:88) p> ˙ f k κ p (cid:0) κ ( ∇ h kp ) + κ ( ∇ h kp ) (cid:1) = κ α ( ∇ f ) f + κ α ( ∇ f ) f + n (cid:88) k,p =3 ˙ f k κ p (cid:0) κ ( ∇ h kp ) + κ ( ∇ h kp ) (cid:1) − κ (cid:32) ˙ f κ ( ∇ h ) + ˙ f κ ( ∇ h ) + ˙ f κ ( ∇ h ) + ˙ f κ ( ∇ h ) (cid:33) − κ (cid:32) ˙ f κ ( ∇ h ) + ˙ f κ ( ∇ h ) + ˙ f κ ( ∇ h ) + ˙ f κ ( ∇ h ) (cid:33) + 2 ˙ f (cid:0) −∇ h ∇ h + ( ∇ h ) (cid:1) + 2 ˙ f (cid:0) −∇ h ∇ h + ( ∇ h ) (cid:1) + 2 ˙ f (cid:88) p> (cid:18) κ κ p ( ∇ h p ) + κ κ p ( ∇ h p ) (cid:19) + 2 ˙ f (cid:88) p> (cid:18) κ κ p ( ∇ h p ) + κ κ p ( ∇ h p ) (cid:19) + 2 (cid:88) k> ˙ f k (cid:0) −∇ k h ∇ k h + ( ∇ k h ) (cid:1) . (3.5)Since ( x , O ) is a minimum point of G at time t , we have ∇ i G = 0 for i = 1 , · · · , n , so κ ∇ i h + κ ∇ i h = 0 , i = 1 , · · · , n. (3.6)Substituting (3.6) into (3.5), the second to the fourth lines on the right of (3.5) vanish,the last line becomes 2 (cid:80) k> ˙ f k (cid:16) κ κ ( ∇ k h ) + ( ∇ k h ) (cid:17) ≥ 0, and the remaining termsare non-negative.We conclude that ∂∂t G ≥ − CG at a spatial minimum point, and hence by the maximumprinciple [20, Lemma 3.5] we have G ≥ e − Ct inf t =0 G > (cid:3) Proof of Theorem 1.2 In this section, we will give the proof of Theorem 1.2. Shape estimate. First, we show that the preservation of the volume of Ω t , togetherwith a reflection argument, implies that the inner radius and outer radius of Ω t are uni-formly bounded from above and below by positive constants. Lemma 4.1. Denote ρ − ( t ) , ρ + ( t ) be the inner radius and outer radius of Ω t , the domainenclosed by M t . Then there exist positive constants c , c depending only on n, M suchthat < c ≤ ρ − ( t ) ≤ ρ + ( t ) ≤ c (4.1) for all time t ∈ [0 , T ) .Proof. We first use the Alexandrov reflection method to estimate the diameter of Ω t . In[10], the first and the third authors have already used the Alexandrov reflection methodin the proof of convergence of the flow. Let γ be a geodesic line in H n +1 , and let H γ ( s ) bethe totally geodesic hyperbolic n -plane in H n +1 which is perpendicular to γ at γ ( s ) , s ∈ R .We use the notation H + s and H − s for the half-spaces in H n +1 determined by H γ ( s ) : H + s := (cid:91) s (cid:48) ≥ s H γ ( s (cid:48) ) , H − s := (cid:91) s (cid:48) ≤ s H γ ( s (cid:48) ) For a bounded domain Ω in H n +1 , denote Ω + ( s ) = Ω ∩ H + s and Ω − ( s ) = Ω ∩ H − s . Thereflection map across H γ ( s ) is denoted by R γ,s . We define S + γ (Ω) := inf { s ∈ R | R γ,s (Ω + ( s )) ⊂ Ω − ( s ) } . It has been proved in [10] that for any geodesic line γ in H n +1 , S + γ (Ω t ) is strictly decreasingalong the flow (1.2) unless R γ, ¯ s (Ω t ) = Ω t for some ¯ s ∈ R . Note that to prove this property,we only need the convexity of the evolving hypersurface M t = ∂ Ω t which is guaranteed bythe positivity of the sectional curvature. The readers may refer to [12, 13] for more detailson the Alexandrov reflection method.Choose R > is contained in some geodesic ball B R ( p )of radius R and centered at some point p in the hyperbolic space. The above reflectionproperty implies that Ω t ∩ B R ( p ) (cid:54) = ∅ for any t ∈ [0 , T ). If not, there exists some time t such that Ω t does not intersect the geodesic ball B R . Choose a geodesic line γ ( s ) withthe property that there exists a geodesic hyperplace Π = H γ ( s ) which is perpendicular to γ ( s ) and is tangent to the geodesic sphere ∂B R , and the domain Ω t lies in the half-space H + s . Then R γ,s (Ω +0 ) = ∅ ⊂ Ω − . Since S + γ (Ω t ) is decreasing, we have R γ,s (Ω + t ) ⊂ Ω − t .However, this is not possible because Ω − t = Ω t ∩ H − s = ∅ and R γ,s (Ω + t ) is obviously notempty.For any t ∈ [0 , T ), let x , x be points on M t = ∂ Ω t such that d ( p, x ) = min { d ( p, x ) : x ∈ M t } and d ( p, x ) = max { d ( p, x ) : x ∈ M t } , where d ( · , · ) is the distance in thehyperbolic space. Since Ω is contained in the geodesic ball B R ( p ) and Ω t ∩ B R ( p ) (cid:54) = ∅ ,we deduce from | Ω t | = | Ω | that x ∈ B R ( p ). If x ∈ B R ( p ), then the diameter of Ω t isbounded from above by R . Therefore it suffices to study the case x / ∈ B R ( p ). Let γ ( s ) bethe geodesic line passing through x and x , i.e., there are numbers s < s ∈ R such that γ ( s ) = x and γ ( s ) = x . We choose the geodesic plane Π = H γ ( s ) for some number s ∈ ( s , s ) such that Π is perpendicular to γ and is tangent to the boundary of B R ( p ) OLUME PRESERVING FLOW 19 Figure 1. Ω t can not leave out B R at p (cid:48) ∈ ∂B R ( p ). Let q = γ ( s ) be intersection point γ ∩ Π. By the Alexandrov reflectionproperty, d ( x , q ) ≤ d ( q, x ). Then the triangle inequality implies d ( p, x ) ≤ d ( p, x ) + d ( x , x ) ≤ d ( p, x ) + 2 d ( q, x ) ≤ d ( p, x ) + 2( d ( q, p (cid:48) ) + d ( p (cid:48) , p ) + d ( p, x )) ≤ R, where we used the fact x ∈ B R ( p ). This shows that the diameter of Ω t is uniformlybounded along the flow (1.2).To estimate the lower bound of the inner radius of Ω t , we project the domain Ω t in the hyperbolic space H n +1 to the unit ball in Euclidean space R n +1 as in [10, § R ,n +1 the Minkowski spacetime, that is the vector space R n +2 endowed withthe Minkowski spacetime metric (cid:104)· , ·(cid:105) given by (cid:104) X, X (cid:105) = − X + (cid:80) ni =1 X i for any vector X = ( X , X , · · · , X n ) ∈ R n +2 . Then the hyperbolic space is characterized as H n +1 = { X ∈ R ,n +1 , (cid:104) X, X (cid:105) = − , X > } An embedding X : M n → H n +1 induces an embedding Y : M n → B (0) ⊂ R n +1 by X = (1 , Y ) (cid:112) − | Y | . Figure 2. Diameter of Ω t is boundedThe induced metrics g Xij and g Yij of X ( M n ) ⊂ H n +1 and Y ( M n ) ⊂ R n +1 are related by g Xij = 11 − | Y | (cid:18) g Yij + (cid:104) Y, ∂ i Y (cid:105)(cid:104) Y, ∂ j Y (cid:105) (1 − | Y | ) (cid:19) Let ˜Ω t ⊂ B (0) be the corresponding image of Ω t in B (0) ⊂ R n +1 , and observe that˜Ω t is a convex Euclidean domain. Then the diameter bound of Ω t implies the diameterbound on ˜Ω t . In particular, | Y | ≤ C < C . This implies that theinduced metrics g Xij and g Yij are comparable. So the volume of ˜Ω t is also bounded belowby a constant depending on the volume of Ω and the diameter of Ω t . Let ω min ( ˜Ω t ) bethe minimal width of ˜Ω t . Then the volume of ˜Ω t is bounded by the a constant timesthe ω min ( ˜Ω t )(diam( ˜Ω t ) n , since ˜Ω t is contained in a spherical prism of height ω min ( ˜Ω t ) andradius diam( ˜Ω t ). It follows that ω min ( ˜Ω t ) is bounded from below by a positive constant C . Since ˜Ω t is strictly convex, an estimate of Steinhagen [29] implies that the inner radius˜ ρ − ( t ) of ˜Ω t is bounded below by ˜ ρ − ( t ) ≥ c ( n ) ω min ≥ C > 0, from which we obtain theuniform positive lower bound on the inner radius ρ − ( t ) of Ω t . This finishes the proof. (cid:3) By (4.1), the inner radius of Ω t is bounded below by a positive constant c . This impliesthat there exists a geodesic ball of radius c contained in Ω t for each t ∈ [0 , T ). The sameargument as in [10, Lemma 4.2] yields the existence of a geodesic ball with fixed centerenclosed by the flow hypersurface on a suitable fixed time interval. OLUME PRESERVING FLOW 21 Lemma 4.2. Let M t be a smooth solution of the flow (1.2) on [0 , T ) with the global term φ ( t ) given by (1.3) . For any t ∈ [0 , T ) , let B ρ ( p ) be the inball of Ω t , where ρ = ρ − ( t ) .Then B ρ / ( p ) ⊂ Ω t , t ∈ [ t , min { T, t + τ } ) (4.2) for some τ depending only on n, α, Ω . Upper bound of F . Now we can use the technique of Tso [30] as in [10] to provethe upper bound of F along the flow (1.2) provided that F satisfies Assumption 1.1. Theinequality (1.4) and the fact that each M t has positive sectional curvature are crucial inthe proof. Theorem 4.3. Assume that F satisfies the Assumption 1.1. Then along the flow (1.2) with any α > , we have F ≤ C for any t ∈ [0 , T ) , where C depends on n, α, M but noton T .Proof. For any given t ∈ [0 , T ), let B ρ ( p ) be the inball of Ω t , where ρ = ρ − ( t ).Consider the support function u ( x, t ) = sinh r p ( x ) (cid:104) ∂r p , ν (cid:105) of M t with respect to thepoint p , where r p ( x ) is the distance function in H n +1 from the point p . Since M t isstrictly convex, by (4.2), u ( x, t ) ≥ sinh( ρ c (4.3)on M t for any t ∈ [ t , min { T, t + τ } ). On the other hand, the estimate (4.1) implies that u ( x, t ) ≤ sinh(2 c ) on M t for all t ∈ [ t , min { T, t + τ } ). Recall that the support function u ( x, t ) evolves by ∂∂t u = ˙Ψ kl ∇ k ∇ l u + cosh r p ( x ) (cid:16) φ ( t ) − Ψ − ˙Ψ kl h kl (cid:17) + ˙Ψ ij h ki h kj u. (4.4)as we computed in [10], where Ψ = F α ( W ). Define the auxiliary function W ( x, t ) = Ψ( x, t ) u ( x, t ) − c , which is well-defined on M t for all t ∈ [ t , min { T, t + τ } ). Combining (2.12) and (4.4),we have ∂∂t W = ˙Ψ ij (cid:18) ∇ j ∇ i W + 2 u − c ∇ i u ∇ j W (cid:19) − φ ( t ) u − c (cid:16) ˙Ψ ij ( h ki h jk − δ ji ) + W cosh r p ( x ) (cid:17) + Ψ( u − c ) (Ψ + ˙Ψ kl h kl ) cosh r p ( x ) − c Ψ( u − c ) ˙Ψ ij h ki h jk − W ˙Ψ ij δ ji . By homogeneity of Ψ and the inverse-concavity of F , we have Ψ + ˙Ψ kl h kl = (1 + α )Ψ and˙Ψ ij h ki h jk ≥ αF α +1 . Moreover, by the inequality (1.4) and the fact that κ κ > 1, we have˙Ψ ij ( h ki h jk − δ ji ) = αf α − n (cid:88) i =1 ˙ f i ( κ i − ≥ αf α − (cid:16) ˙ f ( κ − 1) + ˙ f ( κ − (cid:17) ≥ αf α − ˙ f (cid:18) κ κ ( κ − 1) + ( κ − (cid:19) = αf α − ˙ f κ − ( κ κ − κ + κ ) ≥ , where we used κ i ≥ i = 2 , · · · , n in the first inequality. Then we arrive at ∂∂t W ≤ ˙Ψ ij (cid:18) ∇ j ∇ i W + 2 u − c ∇ i u ∇ j W (cid:19) + ( α + 1) W cosh r p ( x ) − αcW F. (4.5)The remaining proof of Theorem 4.3 is the same with [10, § r p ( x ) ≤ c , we obtain from(4.5) that ∂∂t W ≤ ˙Ψ ij (cid:18) ∇ j ∇ i W + 2 u − c ∇ i u ∇ j W (cid:19) + W (cid:16) ( α + 1) cosh(2 c ) − αc α W /α (cid:17) (4.6)holds on [ t , min { T, t + τ } ). Let ˜ W ( t ) = sup M t W ( · , t ). Then (4.6) implies that ddt ˜ W ( t ) ≤ ˜ W (cid:16) ( α + 1) cosh(2 c ) − αc α ˜ W /α (cid:17) from which it follows from the maximum principle that˜ W ( t ) ≤ max (cid:40)(cid:18) α ) cosh(2 c ) α (cid:19) α c − ( α +1) , (cid:18) 21 + α (cid:19) α α c − ( t − t ) − α α (cid:41) . (4.7)Then the upper bound on F follows from (4.7) and the facts that c = 12 sinh( ρ ≥ 12 sinh( c u − c ≤ c , where c , c are constants in (4.1) depending only on n, M . (cid:3) Long time existence and convergence. In this subsection, we complete the proofof Theorem 1.2. Firstly, in the case (i) of Theorem 1.2, we can deduce directly a uniformestimate on the principal curvatures of M t . In fact, since f ( κ ) is bounded from above byTheorem 4.3, C ≥ f ( κ , κ , · · · , κ n ) ≥ f ( κ , κ , · · · , κ n ) , (4.8)where in the second inequality we used the facts that f is increasing in each argumentand κ i κ > i = 2 , · · · , n . Combining (4.8) and (1.6) gives that κ ≥ C > C . Since the dual function f ∗ of f vanishes on the boundary ofthe positive cone Γ + and f ∗ ( τ i ) = 1 /f ( κ i ) ≥ C > τ i = 1 /κ i ≤ C gives the lower bound on τ i , which is equivalent to the upper bound of κ i .In summary, we obtain uniform two-sided positive bound on the principal curvatures of M t along the flow (1.2) in the case (i) of Theorem 1.2.The examples of f satisfying Assumption 1.1 and the condition (i) of Theorem 1.2includea). n ≥ f = n − /k S /kk with k > OLUME PRESERVING FLOW 23 b). n ≥ f = E /kk with k = 1 , · · · , n ;c). n = 2 , f = ( κ + κ ) / n = 2 , f = ( κ κ ) / . In general,the estimate 1 ≤ κ κ = f ( κ ) ≤ C can not prevent κ from going to infinity. Instead, wewill prove that the pinching ratio κ /κ is bounded from above along the flow (1.2) with f = ( κ κ ) / and α ∈ [1 / , ≤ κ κ ≤ C yields theuniform estimate on κ and κ . Lemma 4.4. In the case n = 2 , f = ( κ κ ) / and α ∈ [1 / , , the principal curvatures κ , κ of M t satisfy < C ≤ κ ≤ κ ≤ C, ∀ t ∈ [0 , T ) (4.9) for some positive constant C along the flow (1.2) .Proof. In this case, Ψ( W ) = ψ ( κ ) = K α/ , where K = κ κ is the Gauss curvature. Thederivatives of ψ with respect to κ i are listed in the following:˙ ψ = α K α − κ , ˙ ψ = α K α − κ (4.10)¨ ψ = α α − K α − κ , ¨ ψ = α α − K α − κ (4.11)¨ ψ = ¨ ψ = α K α − . (4.12)Then we have ˙Ψ ij h ki h jk = n (cid:88) i =1 ˙ ψ i κ i = α K α/ H (4.13)˙Ψ ij δ ji = n (cid:88) i =1 ˙ ψ i = α K α − H, ˙Ψ ij h ji = αK α/ , (4.14)where H = κ + κ is the mean curvature.To prove the estimate (4.9), we define a function G = K α − ( H − K )and aim to prove that G ( x, t ) ≤ max M G ( x, 0) by maximum principle. Using (4.13) and(4.14), the evolution equations (2.12) and (2.13) imply that ∂∂t K = ˙Ψ kl ∇ k ∇ l K + ( α − K − ˙Ψ kl ∇ k K ∇ l K + ( K α/ − φ ( t ))( K − H. (4.15)and ∂∂t H = ˙Ψ kl ∇ k ∇ l H + ¨Ψ kl,pq ∇ i h kl ∇ i h pq + K α − ( K + α − k ))( H − K ) + 2 K α/ ( K − − φ ( t )( H − K − . (4.16) By a direct computation using (4.15) and (4.16), we obtain the evolution equation of G as follows: ∂∂t G = ˙Ψ kl ∇ k ∇ l G − α − K − ˙Ψ kl ∇ k K ∇ l G + ( α − α − K α − ˙Ψ kl ∇ k K ∇ l K ( H − K ) − K α − ˙Ψ kl ∇ k H ∇ l H − α − K α − ˙Ψ kl ∇ k K ∇ l K + 2 HK α − ¨Ψ kl,pq ∇ i h kl ∇ i h pq + 2 HK α − ( H − K ) − HK α − ( H − K )( αK + 2 − α ) φ ( t ) (4.17)We will apply the maximum principle to prove that max M t G is non-increasing in timealong the flow (1.2). We first look at the zero order terms of (4.17), i.e., the terms in thelast line of (4.17) which we denote by Q . Since K = κ κ > φ ( t ) = 1 | M t | (cid:90) M t K α/ ≥ , and αK + 2 − α > . We also have H − K = ( κ − κ ) ≥ 0. Then Q ≤ HK α − ( H − K ) − HK α − ( H − K )( αK + 2 − α )= HK α − ( H − K ) (cid:18) K α/ − αK + α − (cid:19) . For any K > 1, denote f ( α ) = 2 K α/ − αK + α − 2. Then f (2) = f (0) = 0 and f ( α ) is aconvex function of α . Therefore f ( α ) ≤ Q ≤ α ∈ [0 , G , we have ∇ i G = 0 for all i = 1 , 2, which is equivalent to2 H ∇ i H = (cid:0) α − − ( α − K − H (cid:1) ∇ i K. (4.18)Then the gradient terms (denoted by Q ) of (4.17) at the critical point of G satisfy Q K − α = (cid:18) − α − KH − α − α − 2) + ( α − α − H K (cid:19) × ˙Ψ kl ∇ k K ∇ l K + 2 HK ¨Ψ kl,pq ∇ i h kl ∇ i h pq . (4.19)Using (4.10) – (4.12), we have˙Ψ kl ∇ k K ∇ l K = α K α − ( κ ( ∇ K ) + κ ( ∇ K ) )¨Ψ kl,pq ∇ i h kl ∇ i h pq = α α − K α − (cid:88) i =1 ( κ ( ∇ i h ) + κ ( ∇ i h ) )+ α K α − (cid:88) i =1 ∇ i h ∇ i h − αK α − (cid:88) i =1 ( ∇ i h ) = α α − K α − (( ∇ K ) + ( ∇ K ) )+ αK α − (cid:18) ∇ h ∇ h − ( ∇ h ) + ∇ h ∇ h − ( ∇ h ) (cid:19) OLUME PRESERVING FLOW 25 The equation (4.18) implies that ∇ i h and ∇ i h are linearly dependent, i.e., there existfunctions g , g such that g ∇ i h = g ∇ i h . (4.20)The functions g , g can be expressed explicitly as follows: g =2 HK − (4( α − K + (2 − α ) H ) κ ,g = − HK + (4( α − K + (2 − α ) H ) κ . Without loss of generality, we can assume that both g and g are not equal to zero at thecritical point of G . In fact, if g = 0, then0 = g = (cid:18) ( α − H − K ) + 2 Hκ − K (cid:19) κ = (cid:18) ( α − κ − κ ) + 2 κ (cid:19) ( κ − κ ) κ . (4.21)Since α ≤ 2, we have ( α − κ − κ ) + 2 κ ≥ κ > 0. Thus (4.21) is equivalent to κ = κ and we have nothing to prove.By the equation (4.20), we have( ∇ K ) =( κ + g − g κ ) ( ∇ h ) = g − H K ( H − K )( ∇ h ) , ( ∇ K ) =( κ g − g + κ ) ( ∇ h ) = g − H K ( H − K )( ∇ h ) . Using the Codazzi identity, the equation (4.20) also implies that ∇ h ∇ h − ( ∇ h ) + ∇ h ∇ h − ( ∇ h ) = ∇ h ∇ h − ( ∇ h ) + ∇ h ∇ h − ( ∇ h ) = g − g ( g − g )( ∇ h ) + g − g ( g − g )( ∇ h ) =(2 − α )( H − K ) (cid:18) g − g ( ∇ h ) + g − g ( ∇ h ) (cid:19) . Therefore we can write the right hand side of (4.19) as linear combination of ( ∇ h ) and ( ∇ h ) : 2 α Q K − α/ = q g − ( ∇ h ) + q g − ( ∇ h ) , (4.22)where the coefficients q , q satisfy q = (cid:18) − α − K H + 2( α − α − K + ( α − H (cid:19) H K ( H − K ) κ + 4(2 − α ) H K ( H − K )= − α − K ( H − K ) κ − − α )(2 α − H K ( H − K ) κ − − α ) H K ( H − K ) κ and q = − α − K ( H − K ) κ − − α )(2 α − H K ( H − K ) κ − − α ) H K ( H − K ) κ . It can be checked directly that q and q are both non-positive if α ∈ [1 / , Q of (4.17) are non-positive at a critical point of G if α ∈ [1 / , M t G is non-increasing in time. It follows that G ( x, t ) ≤ max M G ( x, < K ≤ C for some constant C > H = 4 K + K α − G ≤ C. (4.23)Finally, the estimate (4.9) follows from (4.23) and K > (cid:3) Now we have proved that the principal curvatures κ i of M t satisfy the uniform estimate0 < /C ≤ κ i ≤ C for some constant C > 0, which is equivalent to the C estimate for M t . Since the functions f we considered in Theorem 1.2 are inverse-concave, we can applyan argument similar to that in [10, § 5] to derive higher regularity estimates. The standardcontinuation argument then implies the long time existence of the flow, and the argumentin [10, § 6] implies the smooth convergence to a geodesic sphere as time goes to infinity.5. Horospherically convex regions In this section we will investigate some of the properties of horospherically convex re-gions in hyperbolic space (that is, regions which are given by the intersection of a collectionof horo-balls). In particular, for such regions we define a horospherical Gauss map, whichis a map to the unit sphere, and we show that each horospherically convex region is com-pletely described in terms of a scalar function on the sphere which we call the horosphericalsupport function . There are interesting formal similarities between this situation and thatof convex Euclidean bodies. For the purposes of this paper the main result we need is thatthe modified Quermassintegrals are monotone with respect to inclusion for horospheri-cally convex domains. However we expect that the description of horospherically convexregions which we develop here will be useful in further investigations beyond the scope ofthis paper.We remark that a similar development is presented in [15], but in a slightly differentcontext: In that paper the ‘horospherically convex’ regions are those which are intersec-tions of complements of horo-balls (corresponding to principal curvatures greater than − The horospherical Gauss map. The horospheres in hyperbolic space are the sub-manifolds with constant principal curvatures equal to 1 everywhere. If we identify H n +1 with the future time-like hyperboloid in Minkowski space R n +1 , , then the condition ofconstant principal curvatures equal to 1 implies that the null vector ¯ e := X − ν is constanton the hypersurface, since we have W = I, and hence D v ¯ e = D v ( X − ν ) = DX ((I − W )( v )) = 0for all tangent vectors v . Then we observe that X · ¯ e = X · ( X − ν ) = − , OLUME PRESERVING FLOW 27 from which it follows that the horosphere is the intersection of the null hyperplane { X : X · ¯ e = − } with the hyperboloid H n +1 . The horospheres are therefore in one-to-onecorrespondence with points ¯ e in the future null cone, which are given by { ¯ e = λ ( e , 1) : e ∈ S n , λ > } , and there is a one-parameter family of these for each e ∈ S n . Forconvenience we parametrise these by their signed geodesic distance s from the ‘north pole’ N = (0 , ∈ H n +1 , satisfying − λ (cosh( s ) N + sinh( s )( e , · ( e , 1) = − λ e s . It followsthat λ = e − s . Thus we denote by H e ( s ) the horosphere { X ∈ H n +1 : X · ( e , 1) = − e s } .The interior region (called a horo-ball ) is denoted by B e ( s ) = { X ∈ H n +1 : 0 > X · ( e , > − e s } . A region Ω in H n +1 is horospherically convex (or h-convex for convenience) if everyboundary point p of ∂ Ω has a supporting horo-ball, i.e. a horo-ball B such that Ω ⊂ B and p ∈ ∂B . If the boundary of Ω is a smooth hypersurface, then this implies that everyprincipal curvature of ∂ Ω is greater than or equal to 1 at p . We say that Ω is uniformlyh-convex if there is δ > δ .Let M n = ∂ Ω be a hypersurface which is at the boundary of a horospherically convexregion Ω. Then the horospherical Gauss map e : M → S n assigns to each p ∈ M the point e ( p ) = π ( X ( p ) − ν ( p )) ∈ S n , where π ( x, y ) = xy is the radial projection from the futurenull cone onto the sphere S n × { } . We observe that the derivative of e is non-singular if M is uniformly h-convex: If v is a tangent vector to M , then D e ( v ) = Dπ (cid:12)(cid:12) X − ν (( W − I)( v )) . Here ˜ v = ( W − I)( v ) is a non-zero tangent vector to M since the eigenvalues κ i of W aregreater than 1. In particular ˜ v is spacelike. On the other hand the kernel of Dπ | X − ν is theline R ( X − ν ) consisting of null vectors. Therefore Dπ (˜ v ) (cid:54) = 0. Thus D e is an injectivelinear map, hence an isomorphism. It follows that e is a diffeomorphism from M to S n .5.2. The horospherical support function. Let M n = ∂ Ω be the boundary of a com-pact h-convex region. Then for each e ∈ S n we define the horospherical support function of Ω (or M ) in direction e by u ( e ) := inf { s ∈ R : Ω ⊂ B e ( s ) } . Alternatively, define f e : H n +1 → R by f e ( ξ ) = log ( − ξ · ( e , H n +1 , and we have the alternative characterisation u ( e ) = sup { f e ( ξ ) : ξ ∈ Ω } . (5.1)The function u is called the horospherical support function of the region Ω, and B e ( u ( e ))is the supporting horo-ball in direction e . The support function completely determines ahorospherically convex region Ω, as an intersection of horo-balls:Ω = (cid:92) e ∈ S n B e ( u ( e )) . (5.2) Recovering the region from the support function. If the region is uniformlyh-convex, in the sense that all principal curvatures are greater than 1, then there is aunique point of M in the boundary of the supporting horo-ball B e ( u ( e )). We denote thispoint by ¯ X ( e ). We observe that ¯ X = X ◦ e − , so if M is smooth and uniformly h-convex(so that e is a diffeomorphism) then ¯ X is a smooth embedding.We will show that ¯ X can be written in terms of the support function u , as follows:Choose local coordinates { x i } for S n near e . We write ¯ X ( e ) as a linear combination of thebasis consisting of the two null elements ( e , 1) and ( − e , e j , e j = ∂ e ∂x j for j = 1 , · · · , n : ¯ X ( e ) = α ( − e , 1) + β ( e , 1) + γ j ( e j , α , β , γ j . Since ¯ X ( e ) ∈ H n +1 we have | γ | − αβ = − 1, so that β = | γ | α . We also know that ¯ X ( e ) · ( e , 1) = − e u ( e ) since ¯ X ( e ) ∈ H e ( u ( e )), implyingthat α = e u . This gives¯ X ( e ) = 12 e u ( e ) ( − e , 1) + 12 e − u ( e ) (1 + | γ | )( e , 1) + γ j ( e j , . Furthermore, the normal to M at the point ¯ X ( e ) must coincide with the normal to thehorosphere H e ( u ( e )), which is given by ν = ¯ X − ¯ e = ¯ X − e − u ( e ) ( e , . (5.3)Since | ¯ X | = − ∂ j ¯ X · ¯ X = 0, and hence0 = ∂ j ¯ X · ν = ∂ j ¯ X · (cid:0) ¯ X − e − u ( e , (cid:1) = − e − u ∂ j X · ( e , . Observing that ( e , · ( e , 1) = 0 and ( e i , · ( e , 1) = 0, and that ∂ j e i = − ¯ g ij e and ∂ j e = e j ,the condition becomes 0 = ∂ j X · ( e , (cid:18) 12 e u u j ( − e , − γ j ( e , (cid:19) · ( e , − e u u j − γ j , where γ j = γ i ¯ g ij and ¯ g is the standard metric on S n . It follows that we must have γ j = − e u u j . This gives the following expression for ¯ X :¯ X ( e ) = (cid:18) − e u ¯ ∇ u + (cid:18) 12 e u | ¯ ∇ u | − sinh u (cid:19) e , 12 e u | ¯ ∇ u | + cosh u (cid:19) (5.4)= − e u u p ¯ g pg ( e q , 0) + 12 (cid:0) e u | ¯ ∇ u | + e − u (cid:1) ( e , 1) + 12 e u ( − e , . (5.5) OLUME PRESERVING FLOW 29 A condition for horospherical convexity. Given a smooth function u , we canuse the expression (5.4) to define a map to hyperbolic space. In this section we determinewhen the resulting map is an embedding defining a horospherically convex hypersurface.If order for ¯ X to be an immersion, we require the derivatives ∂ j ¯ X to be linearly inde-pendent. Since we have constructed ¯ X in such a way that ∂ j X is orthogonal to the normalvector ν to the horosphere B e ( u ( e )), ∂ j ¯ X is a linear combination of the basis for the spaceorthogonal to ν and ¯ X given by the projections E k of ( e k , k = 1 , · · · , n . Computingexplicity, we find E k = ( e k , − u k ( e , . (5.6)The immersion condition is therefore equivalent to invertibility of the matrix A define by A jk = − ∂ j ¯ X · E k . Given that A is non-singular, we have that ¯ X is an immersion with unit normal vector ν ( e ), and we can differentiate the equation X − ν = e − u ( e , 1) to obtain the following: − ( h pj − δ pj ) ∂ p X = − u j e − u ( e , 1) + e − u ( e j , . Taking the inner product with E k using (5.6), we obtain( h pj − δ pj ) A pk = e − u ¯ g jk . (5.7)It follows that A is non-singular precisely when W − I is non-singular, and is given by A jk = e − u (cid:104) ( W − I) − (cid:105) pj ¯ g pk . (5.8)In particular, A is symmetric, and W − I is positive definite (corresponding to uniformh-convexity) if and only if the matrix A is positive definite. We conclude that if u isa smooth function on S n , then the map X defines an embedding to the boundary of auniformly h-convex region if and only if the tensor A computed from u is positive definite.Computing A explicitly using (5.5), we obtain A jk = (cid:18) ( ¯ ∇ j (e u ¯ ∇ u ) , − e u u j ( e , − ∂ j (e u | ¯ ∇ u | + e − u )( e , − 12 e u u j ( − e , − (cid:18) 12 e u |∇ u | − sinh u (cid:19) ( e j , (cid:19) · (( e k , − u k ( e , ∇ j (e u ¯ ∇ k u ) − 12 e u | ¯ ∇ u | ¯ g jk + sinh u ¯ g jk . It is convenient to write this in terms of the function ϕ = e u : A jk = ¯ ∇ j ¯ ∇ k ϕ − | ¯ ∇ ϕ | ϕ ¯ g jk + ϕ − ϕ − g jk . (5.9)5.5. Monotonicity of the modified Quermassintegrals. We will prove that the mod-ified quermassintegrals ˜ W k are monotone with respect to inclusion by making use of thefollowing result: Proposition 5.1. Suppose that Ω ⊂ Ω are smooth, strictly h-convex domains in H n +1 .Then there exists a smooth map X : S n × [0 , → H n +1 such that(1) X ( ., t ) is a uniformly h-convex embedding of S n for each t ; (2) X ( S n , 0) = ∂ Ω and X ( S n , 1) = ∂ Ω ;(3) The hypersurfaces M t = X ( S n , t ) are expanding, in the sense that ∂X∂t · ν ≥ .Equivalently, the enclosed regions Ω t are nested: Ω s ⊂ Ω t for each s ≤ t in [0 , .Proof. Let u and u be the horospherical support functions of Ω and Ω respectively,The inclusion Ω ⊂ Ω implies that u ( e ) ≤ u ( e ) for all e ∈ S n , by the characterisation(5.1).We define X ( e , t ) = ¯ X [ u ( e , t )] according to the formula (5.4), wheree u ( e ,t ) = ϕ ( e , t ) := (1 − t ) ϕ ( e ) + tϕ ( e ) , where ϕ i = e u i for i = 0 , 1. Then u ( e , t ) is increasing in t , and it follows that the regionsΩ t are nested, by the expression (5.2).We check that each Ω t is a strictly h-convex region, by showing that the matrix A constructed from u ( ., t ) is positive definite for each t : We have A jk [ u ( ., t )] = ¯ ∇ j ¯ ∇ k ϕ t − | ¯ ∇ ϕ t | ϕ t ¯ g jk + ϕ t − ϕ − t g jk = (1 − t ) A jk [ u ] + tA jk [ u ]+ 12 (cid:18) − | (1 − t ) ¯ ∇ ϕ + t ¯ ∇ ϕ | (1 − t ) ϕ + tϕ + (1 − t ) | ¯ ∇ ϕ | ϕ + t | ¯ ∇ ϕ | ϕ (cid:19) ¯ g jk + 12 (cid:18) − − t ) ϕ + tϕ + 1 − tϕ + tϕ (cid:19) ¯ g jk = (1 − t ) A jk [ u ] + tA jk [ u ] + t (1 − t ) | ϕ ¯ ∇ ϕ − ϕ ¯ ∇ ϕ | + | ϕ − ϕ | ϕ ϕ ((1 − t ) ϕ + tϕ ) ¯ g jk ≥ (1 − t ) A jk [ u ] + tA jk [ u ] . Since A jk [ u ] and A jk [ u ] are positive definite, so is A jk [ u t ] for each t ∈ [0 , t is uniformly h-convex. (cid:3) Corollary 5.2. The modified quermassintegral (cid:102) W k is monotone with respect to inclusionfor h-convex domains: That is, if Ω and Ω are h-convex domains with Ω ⊂ Ω , then (cid:102) W k (Ω ) ≤ (cid:102) W k (Ω ) .Proof. We use the map X constructed in the Proposition 5.1. By Lemma 2.4 we have ddt (cid:102) W k (Ω t ) = (cid:90) M t E k ( λ ) ∂X∂t · ν dµ t . Since each M t is h-convex, we have λ i > E k ( λ ) > 0, and from Proposition 5.1we have ∂X∂t · ν ≥ 0. It follows that ddt (cid:102) W k (Ω t ) ≥ t , and hence (cid:102) W k (Ω ) ≤ (cid:102) W k (Ω )as claimed. (cid:3) Evolution of the horospherical support function. We end this section with thefollowing observation that the flow (1.17) of h-convex hypersurfaces is equivalent to aninitial value problem for the horospherical support function. OLUME PRESERVING FLOW 31 Proposition 5.3. The flow (1.17) of h-convex hypersurfaces in H n +1 is equivalent to thefollowing initial value problem ∂∂t ϕ = − F (( A ij ) − ) + ϕφ ( t ) ,ϕ ( · , 0) = ϕ ( · ) (5.10) on S n × [0 , T ) , where ϕ = e u and A ij is the matrix defined in (5.9) .Proof. Suppose that X ( · , t ) : M → H n +1 , t ∈ [0 , T ) is a family of smooth, closed andstrictly h-convex hypersurfaces satisfying the flow (1.17). Then as explained in § e is a diffeomorphism from M t = X ( M, t ) to S n . We canreparametrize M t such that ¯ X = X ◦ e − is a family of smooth embeddings from S n to H n +1 . Then ∂∂t ¯ X ( z, t ) = ∂∂t X ( p, t ) + ∂X∂p i ∂p i ∂t , where z ∈ S n and p = e − ( z ) ∈ M t . Since ∂X∂p i is tangent to M t , we have ∂∂t ¯ X ( z, t ) · ν ( z, t ) = ∂∂t X ( p, t ) · ν ( z, t ) = φ ( t ) − F ( W − I ) . (5.11)On the other hand, by (5.3) we have¯ X ( z, t ) − ν ( z, t ) = e − u ( z,t ) ( z, , (5.12)where u ( · , t ) is the horospherical support function of M t and ( z, ∈ R n +1 , is a nullvector. Differentiating (5.12) in time gives that ∂∂t ¯ X ( z, t ) − ∂∂t ν ( z, t ) = − e − u ( z,t ) ∂u∂t ( z, . Then ∂∂t ¯ X ( z, t ) · ν ( z, t ) = − e − u ( z,t ) ∂u∂t ( z, · ν ( z, t )= − e − u ( z,t ) ∂u∂t ( z, · ( ¯ X ( z, t ) − e − u ( z,t ) ( z, − e − u ( z,t ) ∂u∂t ( z, · e u ( z,t ) ( − z, ∂u∂t , (5.13)where we used (5.3) and (5.5). Combining (5.11) and (5.13) implies that ∂u∂t = φ ( t ) − F ( W − I ) . (5.14)Therefore ϕ = e u satisfies ∂ϕ∂t = e u φ ( t ) − F ( e u ( W − I ))= ϕφ ( t ) − F (( A ij ) − ) (5.15)with A ij defined as in (5.9). Conversely, suppose that we have a smooth solution ϕ ( · , t ) of the initial value problem(5.10) with A ij positive definite. Then by the discussion in § X given in(5.5) using the function u = log ϕ defines a family of smooth h-convex hypersurfaces in H n +1 . We claim that we can find a family of diffeomorphisms ξ ( · , t ) : S n → S n such that X ( z, t ) = ¯ X ( ξ ( z, t ) , t ) solves the flow equation (1.17). Since ∂∂t X ( z, t ) = ∂∂t ¯ X ( ξ, t ) + ∂ i ¯ X ∂ξ i ∂t =( ∂∂t ¯ X ( ξ, t ) · ν ( ξ, t )) ν ( ξ, t ) + ( ∂∂t ¯ X ( ξ, t )) (cid:62) + ∂ i ¯ X ∂ξ i ∂t =( φ ( t ) − F ( W − I )) ν ( ξ, t ) + ( ∂∂t ¯ X ( ξ, t )) (cid:62) + ∂ i ¯ X ∂ξ i ∂t , where ( · ) (cid:62) denotes the tangential part, it suffices to find a family of diffeomorphisms ξ : S n → S n such that ( ∂∂t ¯ X ( ξ, t )) (cid:62) + ∂ i ¯ X ∂ξ i ∂t = 0 , which is equivalent to ( ∂∂t ¯ X ( ξ, t )) (cid:62) · E j − A ij ∂ξ i ∂t = 0 . (5.16)By assumption A ij is positive definite on S n × [0 , T ), the standard theory of the ordinarydifferential equations implies that the system (5.16) has a unique smooth solution for theinitial condition ξ ( z, 0) = z . This completes the proof. (cid:3) Proof of Theorem 1.7 In this section, we will give the proof of Theorem 1.7.6.1. Pinching estimate. Firstly, we prove the following pinching estimate for the shiftedprincipal curvatures of the evolving hypersurfaces along the flow (1.17). Proposition 6.1. Let M t be a smooth solution to the flow (1.17) on [0 , T ) and assumethat F satisfies the assumption in Theorem 1.7. Then there exists a constant C > depending only on M such that λ n ≤ Cλ (6.1) for all t ∈ [0 , T ) , where λ n = κ n − is the largest shifted principal curvature and λ = κ − is the smallest shifted principal curvature.Proof. We consider the four cases of F separately.(i). F is concave and F vanishes on the boundary of the positive cone Γ + . Define afunction G = F − tr( S ) on M × [0 , T ). Then the equations (2.14) and (2.16) imply that ∂∂t G = F − ∂∂t tr( S ) − F − tr( S ) ∂∂t F = ˙ F kl ∇ k ∇ l G + 2 F − ˙ F kl ∇ k F ∇ l G + F − n (cid:88) i =1 ¨ F kl,pq ∇ i h kl ∇ i h pq OLUME PRESERVING FLOW 33 + φ ( t ) f − (cid:32) tr( S ) (cid:88) k ˙ f k λ k − f | S | (cid:33) + f − (cid:32) n (cid:88) k ˙ f k λ k − | S | (cid:88) k ˙ f k (cid:33) . (6.2)Since F is concave, by the inequality (2.2) we havetr( S ) (cid:88) k ˙ f k λ k − f | S | = (cid:88) k,l (cid:16) ˙ f k λ k λ l − ˙ f k λ k λ l (cid:17) = 12 (cid:88) k,l ( ˙ f k − ˙ f l )( λ k − λ l ) λ k λ l ≤ , and n (cid:88) k ˙ f k λ k − | S | (cid:88) k ˙ f k = (cid:88) k,l (cid:16) ˙ f k λ k − ˙ f k λ l (cid:17) = 12 (cid:88) k,l ( ˙ f k − ˙ f l )( λ k − λ l ) ≤ . Thus the zero order terms of (6.2) are always non-positive. The concavity of F also impliesthat the third term of (6.2) is non-positive. Then we have ∂∂t G ≤ ˙ F kl ∇ k ∇ l G + 2 F − ˙ F kl ∇ k F ∇ l G. (6.3)The maximum principle implies that the supremum of G over M t is decreasing in timealong the flow (1.17). The assumption that f approaches zero on the boundary of thepositive cone Γ + then guarantees that the region { G ( t ) ≤ sup t =0 G } ⊂ Γ + does not touchthe boundary of Γ + . Since G is homogeneous of degree zero with respect to λ i , this impliesthat λ n ≤ Cλ for some constant C > M for all t ∈ [0 , T ).(ii). F is concave and inverse concave. Define a tensor T ij = S ij − ε tr( S ) δ ji , where ε is chosen such that T ij is positive definite initially. Clearly, 0 < ε ≤ n . The evolutionequation (2.16) implies that ∂∂t T ij = ˙ F kl ∇ k ∇ l T ij + ¨ F kl,pq ∇ i h kl ∇ j h pq − ε (cid:32) n (cid:88) i =1 ¨ F kl,pq ∇ i h kl ∇ i h pq (cid:33) δ ji + (cid:32) n (cid:88) k =1 ˙ f k λ k + 2 f − φ ( t ) (cid:33) T ij − (cid:32) φ ( t ) + n (cid:88) k =1 ˙ f k (cid:33) (cid:16) T ki T kj + 2 ε tr( S ) T ij (cid:17) + ε (cid:32) φ ( t ) + n (cid:88) k =1 ˙ f k (cid:33) (cid:0) | S | − ε (tr( S )) (cid:1) δ ji + n (cid:88) k =1 ˙ f k λ k (1 − εn ) δ ji . (6.4)We will apply the tensor maximum principle in Theorem 2.5 to show that T ij is positivedefinite for t > 0. If not, there exists a first time t > x ∈ M t such that T ij has a null vector v ∈ T x M t , i.e., T ij v j = 0 at ( x , t ). The second line of (6.4) satisfiesthe null vector condition and can be ignored. The last line of (6.4) is also nonnegative,since 0 < ε < n and | S | ≥ (tr( S )) /n . For the gradient terms in (6.4), Theorem 4.1 of [4] implies that ¨ F kl,pq ∇ i h kl ∇ j h pq v i v j − ε (cid:32) n (cid:88) i =1 ¨ F kl,pq ∇ i h kl ∇ i h pq (cid:33) | v | + sup Λ a kl (cid:0) pk ∇ l T ip v i − Λ pk Λ ql T pq (cid:1) ≥ v provided that F is concave and inverse concave. Thus by Theorem2.5, the tensor T ij is positive definite for t ∈ [0 , T ). Equivalently, λ ≥ ε ( λ + · · · + λ n )for any t ∈ [0 , T ), which implies the pinching estimate (6.1).(iii). F is inverse concave and F ∗ approaches zero on the boundary of Γ + . In this case,we define T ij = S ij − εF δ ji , where ε is chosen such that T ij is positive definite initially. By(2.14) and (2.16), ∂∂t T ij = ˙ F kl ∇ k ∇ l T ij + ¨ F kl,pq ∇ i h kl ∇ j h pq + ( ˙ f k λ k + 2 f − φ ( t )) S ij − ( φ ( t ) + n (cid:88) k =1 ˙ f k ) S ik S kj + ˙ f k λ k δ ji − ε ( F − φ ( t )) ˙ f k λ k ( λ k + 2) δ ji . (6.5)Suppose v = e is the null eigenvector of T ij at ( x , t ) for some first time t > 0. Denotethe zero order terms of (6.5) by Q . At the point ( x , t ), εF is the smallest eigenvalue of S ij with corresponding eigenvector v . Then Q v i v j =( ˙ f k λ k + 2 f − φ ( t )) εf | v | + ( f − φ ( t ) − ˙ f k κ k ) ε f | v | + ˙ f k λ k | v | − ε ( f − φ ( t )) ˙ f k λ k ( λ k + 2) | v | = ˙ f k λ k (1 + εφ ( t )) | v | − ε f ( (cid:88) k ˙ f k + φ ( t )) | v | = | v | (cid:18) ˙ f k λ k ε ( λ k − εf ) φ ( t ) + (cid:88) k ˙ f k ( λ k − ε f ) (cid:19) ≥ . By Theorem 2.5, to show that T ij remains positive definite for t > 0, it suffices to showthat Q := ¨ F kl,pq ∇ h kl ∇ h pq + 2 sup Λ ˙ F kl (cid:0) pk ∇ l T p − Λ pk Λ ql T pq (cid:1) ≥ . Note that T = 0 and ∇ k T = 0 at ( x , t ), the supremum over Λ can be computedexactly as follows:2 ˙ F kl (cid:0) pk ∇ l T p − Λ pk Λ ql T pq (cid:1) =2 n (cid:88) k =1 n (cid:88) p =2 ˙ f k (cid:0) pk ∇ k T p − (Λ pk ) T pp (cid:1) =2 n (cid:88) k =1 n (cid:88) p =2 ˙ f k (cid:32) ( ∇ k T p ) T pp − (cid:18) Λ pk − ∇ k T p T pp (cid:19) T pp (cid:33) . OLUME PRESERVING FLOW 35 It follows that the supremum is obtained by choosing Λ pk = ∇ k T p T pp . The required inequalityfor Q becomes: Q = ¨ F kl,pq ∇ h kl ∇ h pq + 2 n (cid:88) k =1 n (cid:88) p =2 ˙ f k ( ∇ k T p ) T pp ≥ . Using (2.1) to express the second derivatives of F and noting that ∇ k T p = ∇ k h p − ε ∇ k F δ p = ∇ k h p at ( x , t ) for p (cid:54) = 1, we have Q = ¨ f kl ∇ h kk ∇ h ll + 2 (cid:88) k>l ˙ f k − ˙ f l λ k − λ l ( ∇ h kl ) + 2 n (cid:88) k =1 n (cid:88) l =2 ˙ f k λ l − εF ( ∇ h kl ) . (6.6)Since f is inverse concave, the inequality (2.3) implies that the first term of (6.6) satisfies¨ f kl ∇ h kk ∇ h ll ≥ f − ( n (cid:88) k =1 ˙ f k ∇ h kk ) − (cid:88) k ˙ f k λ k ( ∇ h kk ) = 2 f − ( ∇ F ) − (cid:88) k ˙ f k λ k ( ∇ h kk ) . Then Q ≥ f − ( ∇ F ) − (cid:88) k ˙ f k λ k ( ∇ h kk ) + 2 (cid:88) k>l ˙ f k − ˙ f l λ k − λ l ( ∇ h kl ) + 2 n (cid:88) k =1 n (cid:88) l =2 ˙ f k λ l − εF ( ∇ h kl ) ≥ f − ( ∇ F ) − f λ ( ∇ h ) − (cid:88) k> ˙ f k λ k ( ∇ h kk ) + 2 (cid:88) k> ˙ f k − ˙ f λ k − λ ( ∇ k h ) − (cid:88) k (cid:54) = l> ˙ f k λ l ( ∇ h kl ) + 2 (cid:88) k> ˙ f λ k − εF ( ∇ k h ) + 2 (cid:88) k> ,l> ˙ f k λ l − εF ( ∇ h kl ) = 2 f − ( ∇ F ) − f λ ( ∇ h ) + 2 (cid:88) k> ˙ f k λ k − λ ( ∇ k h ) + 2 (cid:88) k> ,l> ˙ f k (cid:18) λ l − εF − λ l (cid:19) ( ∇ h kl ) ≥ (cid:32) ε F − ˙ f λ (cid:33) ( ∇ h ) = 2 (cid:32) (cid:80) nk =1 ˙ f k λ k ε F − ˙ f λ (cid:33) ( ∇ h ) ≥ , where we used λ = εF and ∇ k h = ε ∇ k F at ( x , t ), and the inequality in (2.4) dueto the inverse concavity of f . Theorem 2.5 implies that T ij remains positive definite for t ∈ [0 , T ). Equivalently, there holds1 λ ≤ ε f ( λ ) − = 1 ε f ∗ ( 1 λ , · · · , λ n ) (6.7)for all t ∈ [0 , T ). Since f ∗ approaches zero on the boundary of the positive cone Γ + , theestimate (6.7) and Lemma 12 of [9] give the pinching estimate (6.1).(iv). n = 2. In this case, we don’t need any second derivative condition on F . Define G = (cid:18) λ − λ λ + λ (cid:19) . Then G is homogeneous of degree zero of the shifted principal curvatures λ , λ . Theevolution equation (2.16) implies that ∂∂t G = ˙ F kl ∇ k ∇ l G + (cid:16) ˙ G ij ¨ F kl,pq − ˙ F ij ¨ G kl,pq (cid:17) ∇ i S kl ∇ j S pq − (cid:32) φ ( t ) + (cid:88) k ˙ f k (cid:33) ˙ G ij S ik S kj + ( (cid:88) k ˙ f k λ k ) ˙ G ij δ ji . (6.8)The zero order terms of (6.8) are equal to Q = − G λ λ λ + λ (cid:32) φ ( t ) + (cid:88) k ˙ f k (cid:33) − Gλ + λ ( (cid:88) k ˙ f k λ k ) ≤ . The same argument as in [5] gives that the gradient terms of (6.8) are non-positive at thecritical point of G . Then the maximum principles implies that the supremum of G over M t are non-increasing in time along the flow (1.17). This gives the pinching estimate (6.1)and the strict h-convexity of M t for all t ∈ [0 , T ). (cid:3) Shape estimate. Denote by ρ − ( t ) , ρ + ( t ) the inner radius and outer radius of Ω t .Then there exists two points p , p ∈ H n +1 such that B ρ − ( t ) ( p ) ⊂ Ω t ⊂ B ρ + ( t ) ( p ). ByCorollary 5.2, the modified quermassintegral (cid:102) W l is monotone under the inclusion of h-convex domains in H n +1 . This implies that˜ f l ( ρ − ( t )) = (cid:102) W l ( B ρ − ( t ) ( p )) ≤ (cid:102) W l (Ω t ) ≤ (cid:102) W l ( B ρ + ( t ) ( p )) = ˜ f l ( ρ + ( t )) . Along the flow (1.17), (cid:102) W l (Ω t ) = (cid:102) W l (Ω ) is a fixed constant. Therefore, ρ − ( t ) ≤ C ≤ ρ + ( t ) , where C = ˜ f − l ( (cid:102) W l (Ω )) > l, n and Ω .On the other hand, since each Ω t is h-convex , the inner radius and outer radius of Ω t satisfy ρ + ( t ) ≤ c ( ρ − ( t ) + ρ − ( t ) / ) for some uniform positive constant c (see [14, 25]).Thus there exist positive constants c , c depending only on n, l, M such that0 < c ≤ ρ − ( t ) ≤ ρ + ( t ) ≤ c (6.9)for all time t ∈ [0 , T ). OLUME PRESERVING FLOW 37 C estimate.Proposition 6.2. Under the assumptions of Theorem 1.7 with φ ( t ) given in (1.18) , wehave F ≤ C for any t ∈ [0 , T ) , where C depends on n, l, M but not on T .Proof. For any given t ∈ [0 , T ), let B ρ ( p ) be the inball of Ω t , where ρ = ρ − ( t ). Thena similar argument as in [10, Lemma 4.2] yields that B ρ / ( p ) ⊂ Ω t , t ∈ [ t , min { T, t + τ } ) (6.10)for some positive τ depending only on n, l, Ω . Consider the support function u ( x, t ) =sinh r p ( x ) (cid:104) ∂r p , ν (cid:105) of M t with respect to the point p . Then the property (6.10) impliesthat u ( x, t ) ≥ sinh( ρ c (6.11)on M t for any t ∈ [ t , min { T, t + τ } ). On the other hand, the estimate (6.9) implies that u ( x, t ) ≤ sinh(2 c ) on M t for all t ∈ [ t , min { T, t + τ } ). Define the auxiliary function W ( x, t ) = F ( W − I) u ( x, t ) − c on M t for t ∈ [ t , min { T, t + τ } ). Combining (2.14) and the evolution equation (4.4) forthe support function, the function W evolves by ∂∂t W = ˙ F ij (cid:18) ∇ j ∇ i W + 2 u − c ∇ i u ∇ j W (cid:19) − φ ( t ) u − c (cid:16) ˙ F ij ( h ki h jk − δ ji ) + W cosh r p ( x ) (cid:17) + F ( u − c ) ( F + ˙ F kl h kl ) cosh r p ( x ) − cF ( u − c ) ˙ F ij h ki h jk − W ˙ F ij δ ji . (6.12)The second line of (6.12) involves the global term φ ( t ) and is clearly non-positive bythe h-convexity of the evolving hypersurface. By the homogeneity of f with respect to λ i = κ i − 1, we have F + ˙ F kl h kl = 2 F + (cid:80) nk =1 ˙ f k and˙ F ij h ki h jk = ˙ f k ( λ k + 1) = ˙ f k λ k + 2 f + (cid:88) k ˙ f k ≥ Cf , where the last inequality is due to the pinching estimate (6.1). The last term of (6.12) isnon-positive and can be thrown away. In summary, we arrive at ∂∂t W ≤ ˙Ψ ij (cid:18) ∇ j ∇ i W + 2 u − c ∇ i u ∇ j W (cid:19) + W (2 + F − n (cid:88) k =1 ˙ f k ) cosh r p ( x ) − c CW . Note that ˙ f k is homogeneous of degree zero, the pinching estimate (6.1) implies that each˙ f k is bounded from above and below by positive constants. Then without loss of generality we can assume that F − (cid:80) nk =1 ˙ f k ≤ F ≤ (cid:80) nk =1 ˙ f k ≤ C for some constant C > 0. By the upper bound r p ( x ) ≤ c , we obtain the following estimate ∂∂t W ≤ ˙Ψ ij (cid:18) ∇ j ∇ i W + 2 u − c ∇ i u ∇ j W (cid:19) + W (cid:0) c ) − c CW (cid:1) holds on [ t , min { T, t + τ } ). Then the maximum principle implies that W is uniformlybounded from above and the upper bound on F follows by the upper bound on the outerradius in (6.9). (cid:3) Proposition 6.3. There exists a positive constant C , independent of time T , such that F ≥ C > .Proof. Since the evolving hypersurface M t is strictly h-convex, for each time t ∈ [0 , T )there exists a point p ∈ H n +1 and x ∈ M t such that Ω t ⊂ B ρ + ( t ) ( p ) and Ω t ∩ B ρ + ( t ) ( p ) = x . By the estimate (6.9) on the outer radius, the value of F at the point( x , t ) satisfies F ( x , t ) ≥ coth ρ + ( t ) ≥ coth c . Recall that the function F satisfies the evolution equation (2.14) : ∂∂t F = g ik ˙ F ij ∇ k ∇ j F + ( F − φ ( t ))( ˙ F ij h ki h jk − ˙ F ij δ ji ) . (6.13)By the pinching estimate (6.1) and the upper bound on the curvature proved in Proposition6.2, the equation (6.13) is uniformly parabolic and the coefficient of the gradient termsand the lower order terms in (6.13) have bounded C norm. Then there exists r > B r ( x ) × ( t − r , t ] of x and deduce the lower bound F ≥ CF ( x , t ) ≥ C > B r/ ( x ) × ( t − r , t ]. Note that the diameter r of the space-time neighbourhood is not dependent on the point ( x , t ). Consider the boundary point x ∈ ∂B r/ ( x ). We can look at the equation (6.13) in a neighborhood B r ( x ) × ( t − r , t ]of the point ( x , t ). The Harnack inequality implies that F ≥ CF ( x , t ) ≥ C > B r/ ( x ) × ( t − r , t ]. Since the diameter of each M t is uniformly bounded from above,after a finite number of iterations, we conclude that F ≥ C > M t for a uniformconstant C independent of t . (cid:3) The pinching estimate (6.1) together with the bounds on F proven in Proposition 6.2and Proposition 6.3 implies that the shifted principal curvatures λ = ( λ , · · · , λ n ) satisfy0 < C − ≤ λ i ≤ C for some constant C > t ∈ [0 , T ). This gives the uniform C estimate of theevolving hypersurfaces M t . Moreover, the global term φ ( t ) given in (1.18) satisfies 0 Long time existence and convergence. If F is inverse-concave, by applying thesimilar argument as [10, 24] (see also [31]) to the equation (5.10), we can first derive the C ,α estimate and then the C k,α estimate for all k ≥ 2. If F is concave or n = 2, we writethe flow (1.17) as a scalar parabolic PDE for the radial function as follows: Since each OLUME PRESERVING FLOW 39 M t is strictly h-convex, we write M t as a radial graph over a geodesic sphere for a smoothfunction ρ on S n . Let { θ i } , i = 1 , · · · , n be a local coordinate system on S n . The inducedmetric on M t from H n +1 takes the form g ij = ¯ ∇ i ρ ¯ ∇ j ρ + sinh ρ ¯ g ij , where ¯ g ij denotes the round metric on S n . Up to a tangential diffeomorphism, the flowequation (1.17) is equivalent to the following scalar parabolic equation ∂∂t ρ = ( φ ( t ) − F ( W − I)) (cid:113) | ¯ ∇ ρ | / sinh ρ. (6.14)for the smooth function ρ ( · , t ) on S n . The Weingarten matrix W = ( h ji ) can be expressedas h ji = coth ρv δ ji + coth ρv sinh ρ ¯ ∇ j ρ ¯ ∇ i ρ − ˜ σ jk v sinh ρ ¯ ∇ k ¯ ∇ i ρ, where v = (cid:113) | ¯ ∇ ρ | / sinh ρ, and ˜ σ jk = σ jk − ¯ ∇ j ρ ¯ ∇ k ρv sinh ρ . Thus we can apply the argument as in [3, 23] to derive the higher regularity estimate.Therefore, for any F satisfying the assumption of Theorem 1.7, the solution of the flow(1.17) exists for all time t ∈ [0 , ∞ ) and remains smooth and strictly h-convex. Moreover,the Alexandrov reflection argument as in [10, § 6] implies that the flow converges smoothlyas time t → ∞ to a geodesic sphere ∂B r ∞ which satisfies (cid:102) W l ( B r ∞ ) = (cid:102) W l (Ω ). This finishesthe proof of Theorem 1.7.7. Conformal deformation in the conformal class of ¯ g In this section we mention an interesting connection (closely related to the results of[15]) between flows of h-convex hypersurfaces in hyperbolic space by functions of principalcurvatures, and conformal flows of conformally flat metrics on S n . This allows us to trans-late some of our results to convergence theorems for metric flows, and our isoperimetricinequalities to corresponding results for conformally flat metrics.The crucial observation is that there is a correspondence between conformally flat met-rics on S n satisfying a certain curvature inequality, and horospherically convex hyper-surfaces. To describe this, we recall that the isometry group of H n +1 coincides with O + ( n + 1 , S n , by the followingcorrespodence: If L ∈ O + ( n + 1 , ρ L from S n to S n by ρ L ( e ) = π ( L ( e , , where π ( x, y ) = xy is the radial projection from the future null cone to the sphere atheight 1. This defines a group homomorphism from O + ( n + 1 , 1) to the group of M¨obiustransformations. We have the following result: Proposition 7.1. If L ∈ O + ( n + 1 , and M ⊂ H n +1 is a horospherically convex hyper-surface with horospherical support function u : S n → R , denote by u L the horosphericalsupport function of L ( M ) . Then ρ L is an isometry from e − u ¯ g to e − u L ¯ g . That is, e − u ( e ) ¯ g e ( v , v ) = e − u L ( ρ L ( e )) ¯ g ρ L ( e ) ( Dρ L ( v ) , Dρ L ( v )) for all e ∈ S n and v , v ∈ T e S n .Proof. We compute:e − u ( e ) = − X · ( e , − L ( X ) · L ( e , − L ( X ) · µ ( e L , 1) where µ = | L ( e , · (0 , | = µ e − u L ( e L ) . On the other hand the M¨obius transformation ρ L is a conformal transformation withconformal factor µ = | L ( e , · (0 , | . The result follows directly. (cid:3) Corollary 7.2. Isometry invariants of a horospherically convex hypersurface M are M¨obiusinvariants of the conformally flat metric ˜ g = e − u ¯ g , and vice versa. In particular, Rie-mannian invariants of g are isometry invariants of M . Computing explicitly, we find that for n > S ij = 1 n − (cid:32) ˜ R ij − ˜ R n − 1) ˜ g ij (cid:33) of ˜ g (which completely determines the curvature tensor for a conformally flat metric) isgiven by ˜ S ij = 12 ¯ g ij + ¯ ∇ i ¯ ∇ j u + u i u j − | ¯ ∇ u | ¯ g ij = e − u A ij + 12 ˜ g ij = (cid:2) ( W − I) − (cid:3) pi ˜ g pk + 12 ˜ g ij . It follows that the eigenvalues of ˜ S ij (with respect to ˜ g ij ) are + λ i , where λ i = κ i − 1. When n = 2 the tensor ˜ S ij defined by the right-hand side of the above equation isby construction M¨obius-invariant, and so gives a M¨obius invariant of ˜ g which is not aRiemannian invariant. This tensor still has the same relation to the principal curvaturesof the corresponding h-convex hypersurface.We observe that this connection between the Schouten tensor of ˜ g and the Weingartenmap of the hypersurface leads to a conincidence between the corresponding evolutionequations: If a family of h-convex hypersurfaces M t = X ( M, t ) evolves according to acurvature-driven evolution equation of the form ∂X∂t = − F ( W − I , t ) ν OLUME PRESERVING FLOW 41 then the metric ˜ g satisfies ˜ S > ˜ g , and evolves according to the parabolic conformal flow ∂ ˜ g∂t = 2 F (( ˜ S − 12 ˜ g ) − , t )˜ g. In particular the convergence theorems for hypersurface flows correspond to convergencetheorems for the corresponding conformal flows, and the resulting geometric inequalitiesfor hypersurfaces imply corresponding geometric inequalities for the metric ˜ g . References [1] Ben Andrews, Contraction of convex hypersurfaces in Euclidean space , Calc. Var. 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