Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II
aa r X i v : . [ m a t h . DG ] A p r HYPERBOLIC ALEXANDROV-FENCHEL QUERMASSINTEGRALINEQUALITIES II
YUXIN GE, GUOFANG WANG, AND JIE WU
Abstract.
In this paper we first establish an optimal Sobolev type inequality for hypersurfacesin H n (see Theorem 1.1). As an application we obtain hyperbolic Alexandrov-Fenchel inequal-ities for curvature integrals and quermassintegrals. Precisely, we prove a following geometricinequality in the hyperbolic space H n , which is a hyperbolic Alexandrov-Fenchel inequality, Z Σ σ k ≥ C kn − ω n − ((cid:18) | Σ | ω n − (cid:19) k + (cid:18) | Σ | ω n − (cid:19) k n − − kn − ) k , provided that Σ is a horospherical convex, where 2 k ≤ n −
1. Equality holds if and only if Σ is ageodesic sphere in H n . Here σ j = σ j ( κ ) is the j -th mean curvature and κ = ( κ , κ , · · · , κ n − )is the set of the principal curvatures of Σ. Also, an optimal inequality for quermassintegrals in H n is as following: W k +1 (Ω) ≥ ω n − n k X i =0 n − − kn − − k + 2 i C ik (cid:18) nW (Ω) ω n − (cid:19) n − − k +2 in − , provided that Ω ⊂ H n is a domain with Σ = ∂ Ω horospherical convex, where 2 k ≤ n − H n . Here W r (Ω) is quermassintegrals inintegral geometry. Introduction
In this paper we first establish Sobolev type inequalities for hypersurfaces in the hyperbolicspace H n . Let g be a Riemannian metric on a Riemannian manifold. Its k th Gauss-Bonnetcurvature (or Lovelock curvature) L k is defined by (see [13] for instance)(1.1) L k := 12 k δ i i ··· i k − i k j j ··· j k − j k R i i j j · · · R i k − i k j k − j k . Here R ijkl is the Riemannian curvature with respect to g , and the generalized Kronecker deltais defined by δ j j ...j r i i ...i r = det δ j i δ j i · · · δ j r i δ j i δ j i · · · δ j r i ... ... ... ... δ j i r δ j i r · · · δ j r i r . The first named author is partly supported by ANR project ANR-08-BLAN-0335-01. The second and thirdnamed authors are partly supported by SFB/TR71 “Geometric partial differential equations” of DFG.
When k = 1, L is just the scalar curvature R . When k = 2, it is the so-called (second)Gauss-Bonnet curvature L = k Rm k − k Ric k + R , where Rm , Ric are the Riemannian curvature tensor, and the Ricci tensor with respect to g respectively. The Gauss-Bonnet curvature L k is a very natural generalization of the scalarcurvature. When the underlying manifold is local conformally flat, L k equals to the σ k -scalarcurvature up to a constant multiple, precisely(cf. [14]) L k = 2 k k !( n − − k )( n − − k ) · · · ( n − k ) σ k ( g ) . (1.2)Here the σ k -scalar curvature was introduced in Viaclovsky [30] by σ k ( g ) := σ k (Λ g ) , (1.3)and Λ g is the set of the eigenvalues of the Schouten tensor A g defined by(1.4) A g = 1 n − (cid:18) Ric g − R g n − g (cid:19) . Here we consider the ( n − M with metric g . The σ k -scalar curvature isalso a very natural generalization of the scalar curvature R (in fact, σ ( g ) = n − R ) and hasbeen intensively studied in the fully nonlinear Yamabe problem. The fully nonlinear Yamabeproblem for σ k is a generalization of ordinary Yamabe problem for the scalar curvature R . Inthe ordinary Yamabe problem, the following functional, the so-called Yamabe functional, playsa crucial role F ( g ) = ( vol ( g )) − n − n − Z R g dµ ( g ) . (1.5)For a given conformal class [ g ] = { e − u g | u ∈ C ∞ ( M ) } , the Yamabe constant is defined by Y ([ g ]) = inf ˜ g ∈ [ g ] F (˜ g ) . By the resolution of the Yamabe problem, Aubin and Schoen [2, 25] proved that for any metric g on MY ([ g ]) ≤ Y ([ g S n − ]) and Y ([ g ]) < Y ([ g S n − ]) for any ( M, [ g ]) other than [ g S n − ] , (1.6)where [ g S n − ] is the conformal class of the standard round metric on the sphere S n − . From this,one can see the importance of the constant Y ([ g S n − ]). In fact, one can prove that Y ([ g S n − ]) = ( n − n − ω n − n − , (1.7)where ω n − is the volume of g S n − . It is trivial to see that (1.7) is equivalent to Z M L dµ ( g ) = Z M R g dµ ( g ) ≥ ( n − n − ω n − n − vol ( g ) n − n − , (1.8)for any g ∈ [ g S n − ], which is in fact an optimal Sobolev inequality. See [20]. As a naturalgeneralization, we proved in [19] a generalized Sobolev inequality for σ k -scalar curvature σ k ( g ),which states Z M σ k ( g ) dµ ( g ) ≥ C kn − k ω kn − n − vol ( g ) n − − kn − , (1.9) YPERBOLIC ALEXANDROV-FENCHEL INEQUALITIES II 3 for any g ∈ C k − ([ g S n − ]) , where C k − ([ g S n − ]) = [ g S n − ] ∩ Γ + k − and Γ + k = { g | σ j ( g ) > , ∀ j ≤ k } .In this paper, we denote C kn − = ( n − k !( n − − k )! . By (1.2) inequality (1.9) can be written in thefollowing form Z Σ L k dµ ( g ) ≥ C kn − (2 k )! ω kn − n − ( vol ( g )) n − − kn − , (1.10)for any g ∈ C k − ([ g S n − ]) . We call both inequalities (1.8), (1.10) optimal Sobolev inequalities andwould like to investigate which classes of metrics satisfy the optimal Sobolev inequalities. (1.8)and (1.10) mean that a suitable subclass of the conformal class of the standard round metricsatisfies the optimal Sobolev inequalities. From (1.6) we know in any conformal class other thanthe conformal class of the standard round metric, there exist many metrics which do not satisfythe optimal Sobolev inequality. Hence it is natural to ask if there are other interesting classesof metrics satisfy the optimal Sobolev inequality? Observe that for a closed hypersurface Σ in R n , L k = (2 k )! σ k , (1.11)where σ k is the 2 k -mean curvature of Σ, which is defined by σ j = σ j ( κ ) , where κ = ( κ , κ , · · · , κ n − ), κ j (1 ≤ j ≤ n −
1) is the principal curvature of B , and B is the 2ndfundamental form of Σ induced by the standard Euclidean metric. The classical Alexandrov-Fenchel inequality (see [27] for instance) implies for convex hypersurfaces in R n that Z Σ L k ( g ) dµ ( g ) = (2 k )! Z Σ σ k dµ ( g ) ≥ C kn − (2 k )! ω kn − n − | Σ | n − − kn − . (1.12)I this paper we use | Σ | to denote the area of Σ with respect to the induced metric. Inequality(1.12) means that the induced metric of any convex hypersurfaces in R n satisfy the optimalSobolev inequalities. The convexity can be weakened. See the work of Guan-Li [18], Huisken[21] and Chang-Wang [6].In this paper we prove that the induced metric of horospherical convex hypersurfaces in H n also satisfy the optimal Sobolev inequalities. Theorem 1.1.
Let k < n − . Any horospherical convex hypersurfaces Σ in H n satisfies Z Σ L k dµ ( g ) ≥ C kn − (2 k )! ω kn − n − | Σ | n − − kn − , (1.13) equality holds if and only if Σ is a geodesic sphere. A hypersurface in H n is horospherical convex if all principal curvatures are larger than orequal to 1. The horospherical convexity is a natural geometric concept, which is equivalentto the geometric convexity in Riemannian manifolds. For any hypersurface in H n , the Gauss-Bonnet curvature L k of the induced metric of the hypersurface can be expressed in terms of thecurvature integrals by (see also Lemma 3.1 below) L k = C kn − (2 k )! k X j =0 ( − j C jk C k − jn − σ k − j . (1.14) YUXIN GE, GUOFANG WANG, AND JIE WU
Comparing (1.12) for R n with (1.13) for H n and (1.11) with (1.14), we obtain the same inequalityfor L k , while L k has diferent expression in terms of the curvature integrals. We remark thatwhen 2 k = n −
1, (1.13) is an equality for any hypersurface diffeomorphic to a sphere, i.e, Z Σ L n − dv ( g ) = ( n − ω n − . This follows that the Gauss-Bonnet-Chern theorem.As a first direct application, we establish Alexandrov-Fenchel type inequalities for curvatureintegrals.
Theorem 1.2.
Let k ≤ n − . Any horospherical convex hypersurface Σ ⊂ H n satisfies (1.15) Z Σ σ k ≥ C kn − ω n − ((cid:18) | Σ | ω n − (cid:19) k + (cid:18) | Σ | ω n − (cid:19) k n − − kn − ) k , equality holds if and only if Σ is a geodesic sphere. When k = 1 Theorem 1.1, and hence Theroem 1.2, is true even for any star-shaped andtwo-convex hypersurfaces in H n , ie., σ ≥ σ ≥
0, which was proved by Li-Wei-Xiong ina recent work [22]. When k = 2, Theorem 1.1 was proved in our recent paper [15]. Due to thecomplication of the variational structure of R σ k in the hyperbolic space, the case k ≥ k = 1. For case k ≥ σ and propose a conjecturefor general odd σ k +1 .Another application is an optimal inequality for quermassintegrals in H n . For a (geodesically)convex domain Ω ⊂ H n with Σ = ∂ Ω, quermassintegrals are defined by W r (Ω) := ( n − r ) ω r − · · · ω nω n − · · · ω n − r − Z L r χ ( L ∩ Ω) dL, (1.16)where L r is the space of r -dimensional totally geodesic subspaces L in H n , ω r is the area of the r -dimensional standard round sphere and dL is the natural (invariant) measure on L r (cf. [24],[28]). As in the Euclidean case we take W (Ω) = V ol (Ω). With these definitions, unlike theeuclidean case, the quermassintegral in H n do not coincide with the mean curvature integrals,but they are closely related (cf. [28])(1.17)1 C rn − Z Σ σ r = n (cid:18) W r +1 (Ω) + rn − r + 1 W r − (Ω) (cid:19) , W (Ω) = V ol (Ω) , W (Ω) = 1 n | Σ | . The relationship between W and W , the hyperbolic isoperimetric inequality, was establishedby Schmidt [26] 70 years ago. When n = 2, the hyperbolic isoperimetric inequality is L ≥ πA + A , where L is the length of a curve γ in H and A is the area of the enclosed domain by γ . Ingeneral, this hyperbolic isoperimetric inequality has no explicit form. There are many attemptsto establish relationship between W k (Ω) in the hyperbolic space H n . See, for example, [24] YPERBOLIC ALEXANDROV-FENCHEL INEQUALITIES II 5 and [29]. In [11], Gallego-Solanes proved by using integral geometry the following interestinginequality for convex domains in H n , precisely, there holds, W r (Ω) > n − rn − s W s (Ω) , r > s, (1.18)which implies Z Σ σ k dµ > cC kn − | Σ | , (1.19)where c = 1 if k > c = ( n − / ( n −
1) if k = 1 and | Σ | is the area of Σ. Here dµ is thearea element of the induced metric. The constants in (1.18) and (1.19) are optimal in the sensethat one can not replace them by bigger constants. However, they are far away being optimal.As another application of Theorem 1.1, we have the following optimal inequalities of W k (Ω)for general odd k in terms of W = n | Σ | . Theorem 1.3.
Let k ≤ n − . If Ω ⊂ H n be a domain with Σ = ∂ Ω horospherical convex, then (1.20) W k +1 (Ω) ≥ ω n − n k X i =0 n − − kn − − k + 2 i C ik (cid:18) nW (Ω) ω n − (cid:19) n − − k +2 in − , where ω n − is the area of the unit sphere S n − . Equality holds if and only if Σ is a geodesicsphere. As a direct corollary we solve an isoperimetric problem for horospherical convex surfaces withfixed W . Corollary 1.4.
Let k ≤ n − . In a class of horospherical convex hypersurfaces in H n withfixed W , the minimum of W k +1 is achieved by and only by the geodesic spheres. Corollary 1.4 answers a question asked in the paper of Gao, Hug and Schneider [12] in thiscase.In order to prove Theorem 1.1, motivated by [15] and [22] (see also [4] and [9]), we considerthe following functional(1.21) Q (Σ) := | Σ | − n − − kn − Z Σ L k . Here L k is the Gauss-Bonnet curvature with respect to the induced metric g on Σ. This is aYamabe type functional. One of crucial points of this paper is to show that functional Q isnon-increasing under the following inverse curvature flow(1.22) ∂ Σ t ∂t = n − k k σ k − σ k ν, where ν is the outer normal of Σ t , provided that the initial hypersurface is horospherical convex.One can show that horospherical convexity is preserved by flow (1.22). By the convergenceresults of Gerhardt [16] on the inverse curvature flow (1.22), we show that the flow approaches YUXIN GE, GUOFANG WANG, AND JIE WU to surfaces whose induced metrics belong to the conformal class of the standard round spheremetric. Therefore, we can use the result (1.10) to Q (Σ) ≥ lim t →∞ Q (Σ t ) ≥ C kn − (2 k )! ω kn − n − . The rest of this paper is organized as follows. In Section 2 we present some basic factsabout the elementary functions σ k and recall the generalized Sobolev inequality (1.10) from[19]. In Section 3, We present the relationship between various geometric quantities includingthe intrinsic geometric quantities R Σ L k , the curvature integrals R Σ σ k and the quermassintegrals W r (Ω) . In Section 4 we prove the crucial monotonicity of Q and analyze its asymptotic behaviorunder flow (1.22). The proof of our main theorems are given in Section 5. In Section 6, we showthat a similar inequality holds for σ and propose a conjecture for integral integrals σ k +1 .2. Preliminaries
Let σ k be the k -th elementary symmetry function σ k : R n − → R defined by σ k (Λ) = X i < ··· , ∀ j ≤ k } . A symmetric matrix B is called belong to Γ + k if λ ( B ) ∈ Γ + k . We collect the basic facts about σ k , which will be directly used in this paper. For other related facts, see a survey of Guan [17]or [22].(2.1) σ k ( B ) = 1 k ! δ i ··· i k j ··· j k b j i · · · b j k i k , where B = ( b ij ). In the following, for simplicity of notation we denote p k = σ k C kn − . Lemma 2.1.
For Λ ∈ Γ + k , we have the following Newton-MacLaurin inequalities p k − p k +1 p k ≤ , (2.2) p p k − p k ≥ . (2.3) Moreover, equality holds in (2.2) or (2.3) at Λ if and only if Λ = c (1 , , · · · , . YPERBOLIC ALEXANDROV-FENCHEL INEQUALITIES II 7
The Newton-MacLaurin inequalities play a very important role in proving geometric inequal-ities mentioned above. However, we will see that these inequalities are not precise enough toshow our inequality (1.13).Let H n = R + × S n − with the hyperbolic metric¯ g = dr + sinh rg S n − , where g S n − is the standard round metric on the unit sphere S n − and Σ ⊂ H n a smooth closedhypersurface in H n with a unit outward normal ν . Let h be the second fundamental form of Σand κ = ( κ , · · · , κ n − ) the set of principal curvatures of Σ in H n with respect to ν . The k -thmean curvature of Σ is defined by σ k = σ k ( κ ) . We now consider the following curvature evolution equation(2.4) ddt X = F ν, where Σ t = X ( t, · ) is a family of hypersurfaces in H n , ν is the unit outward normal to Σ t = X ( t, · )and F is a speed function which may depend on the position vector X and principal curvaturesof Σ t . One can check that [23] along the flow ddt Z Σ σ k dµ =( k + 1) Z Σ F σ k +1 dµ + ( n − k ) Z Σ F σ k − dµ, (2.5)and thus(2.6) ddt Z Σ p k dµ = Z Σ (cid:0) ( n − k − p k +1 + kp k − (cid:1) F dµ.
If one compares flow (2.4) in H n with a similar flow of hypersurfaces in R n , the last term in (2.5)is an extra term. This extra term comes from −
1, the sectional curvature of H n and makes thephenomenon of H n quite different from the one of R n .As mentioned above we use the following inverse flow(2.7) ddt X = p k − p k ν. By using the result of Gerhardt [16] we have
Proposition 2.2.
If the initial hypersurface Σ is horospherical convex, then the solution forthe flow (2.7) exists for all time t > and preservs the condition of horospherical convexity.Moreover, the hypersurfaces Σ t become more and more umbilical in the sense of | h ij − δ ij | ≤ Ce − tn − , t > , i.e., the principal curvatures are uniformly bounded and converge exponentially fast to one. Here h ij = g ik h kj , where g is the induced metric and h is the second fundamental form.Proof. For the long time existence of the inverse curvature flow, see the work of Gerhardt [16].The preservation of the horospherical convexity along flow (2.7) was proved in [15] with the helpof a maximal principle for tensors of Andrews [1] . (cid:3)
YUXIN GE, GUOFANG WANG, AND JIE WU
Let g be a Riemannian metric on M n − . Denote Ric g and R g the Ricci tensor and the scalarcurvature of g respectively. The Schouten tensor A g is defined by (1.4).The σ k -scalar curvature,which is introduced by Viaclovsky [30], is defined by σ k ( g ) := σ k ( A g ) . This is a natural generalization of the scalar curvature R . In fact, σ ( g ) = n − R . Recall that M is of dimension n −
1. We now consider the conformal class [ g S n − ] of the standard sphere S n − and the following functionals defined by(2.8) F k ( g ) = vol ( g ) − n − − kn − Z S n − σ k ( g ) dµ, k = 0 , , ..., n − . If a metric g satisfies σ j ( g ) > j ≤ k , we call it k -positive and denote g ∈ Γ + k . FromTheorem 1.A in [19] we have Proposition 2.3.
Let < k < n − and g ∈ [ g S n − ] k -positive. We have (2.9) F k ( g ) ≥ F k ( g S n − ) = C kn − k ω kn − n − . Inequality (2.9) is a generalized Sobolev inequality, since when k = 1 inequality (2.9) is justthe optimal Sobolev inequality. See for example [20]. For another Sobolev inequalities, see also[3] and [7]. 3. Relationship between various geometric quantities
The Gauss-Bonnet curvatures L k , and hence R Σ L k are intrinsic geometric quantities, whichdepend only on the induced metric g on Σ and do not depend on the embeddings of (Σ , g ).Lemma 3.2 and Lemma 3.3 below imply that σ k , R σ k and W k +1 are also intrinsic. σ k +1 , R σ k +1 and W k are extrinsic. The functionals R Σ L k are new geometric quantities for thestudy of the integral geometry in H n . In this section we present the relationship between thesegeometric quantities.We first have a relation between L k and σ k . Lemma 3.1.
For a hypersurface (Σ , g ) in H n , its Gauss-Bonnet curvature L k can be expressedby higher order mean curvatures L k = C kn − (2 k )! k X i =0 C ik ( − i p k − i . (3.1) Hence we have Z Σ L k = C kn − (2 k )! k X i =0 C ik ( − i Z Σ p k − i = C kn − (2 k )! k X i =0 ( − i C ik C k − in − Z Σ σ k − i . (3.2) Proof.
First recall the Gauss formula R ijkl = ( h ik h j l − h il h j k ) − ( δ ik δ j l − δ il δ jk ) , YPERBOLIC ALEXANDROV-FENCHEL INEQUALITIES II 9 where h ij := g ik h kj and h is the second fundamental form. Then substituting the Gauss formulaabove into (1.1) and recalling (2.1), we have by a straightforward calculation, L k = 12 k δ i i ··· i k − i k j j ··· j k − j k R i i j j · · · R i k − i k j k − j k = δ i i ··· i k − i k j j ··· j k − j k ( h i j h i j − δ i j δ i j ) · · · ( h i k − j k − h i k j k − δ i k − j k − δ i k j k )= k X i =0 C ik ( − i ( n − k )( n − k + 1) · · · ( n − − k + 2 i ) (cid:0) (2 k − i )! σ k − i (cid:1) = C kn − (2 k )! k X i =0 C ik ( − i p k − i . Here in the second equality we use the symmetry of generalized Kronecker delta and in the thirdequality we use (2.1) and the basic property of generalized Kronecker delta δ i i ··· i p − i p j j ··· j p − j p δ i j = ( n − p ) δ i i ··· i p j j ··· j p , (3.3)which follows from the Laplace expansion of determinant. (cid:3) Motivated by the expression (3.1), we introduce the following notations,(3.4) e L k = k X i =0 C ik ( − i p k − i , e N k = k X i =0 C ik ( − i p k − i +1 . It is clear that L k = (2 k )! C kn − e L k , N k = (2 k )! C kn − e N k . Lemma 3.2.
We have (3.5) σ k = C kn − p k = C kn − (cid:18) k X i =0 C ik e L i (cid:19) , and hence Z Σ σ k = C kn − k X i =0 C ik Z Σ e L i = 1(2 k )! k X i =0 C ik Z Σ L i . To show Theorem 1.3 below, we need
Lemma 3.3.
The quermassintegral W k +1 can be expressed in terms of integral of e L i (3.6) W k +1 (Ω) = 1 n k X i =0 C ik n − − kn − − k + 2 i Z Σ e L k − i . Proof.
We use the induction argument to show (3.6). When k = 0, we have by (1.17) that W ( Q ) = n | Σ | . We then assume that (3.6) holds for k −
1, that is W k − (Ω) = 1 n k − X j =0 C jk − n + 1 − kn + 1 − k + 2 j e L k − − j = 1 n Z Σ k X i =1 C i − k − n + 1 − kn − − k + 2 i e L k − i . (3.7)By (1.17) and (3.5), we have W k +1 (Ω) = 1 n Z Σ p k − kn − k + 1 W k − (Ω)= 1 n Z Σ k X i =0 C ik e L i − kn − k + 1 W k − (Ω) . Substituting (3.7) into above, one immediately obtains (3.6) for k . Thus we complete theproof. (cid:3) One can also show the following relation between the quermassintegrals and the curvatureintegrals.
Lemma 3.4. W k +1 (Ω) = 1 n k X j =0 ( − j (2 k )!!( n − k − k − j )!!( n − k − j )!! 1 C k − jn − Z Σ σ k − j , (3.8) where (2 k − k − k − · · · and (2 k )!! := (2 k )(2 k − · · · . Proof.
One can show this relation by a direct computation. See also [24] or [29]. (cid:3) Monotonicity
In this section we prove the monotonicity of functional Q under inverse curvature flow. First,we have the variational formula for R e L k . Lemma 4.1.
Along the inverse flow (2.7), we have (4.1) ddt Z Σ e L k = ( n − − k ) Z e L k + ( n − − k ) Z Σ (cid:16) e N k p k − p k − e L k (cid:17) . YPERBOLIC ALEXANDROV-FENCHEL INEQUALITIES II 11
Proof.
It follows from (2.6) that along the inverse flow (2.4), we have ddt Z Σ e L k = Z Σ k X i =0 C ik ( − i (cid:16)(cid:0) n − − k + 2 i (cid:1) p k − i +1 + 2( k − i ) p k − i − (cid:17) F = Z Σ k X i =0 C ik ( − i (cid:0) n − − k + 2 i (cid:1) p k − i +1 F + Z k X j =1 C j − k ( − j − k − j + 1) p k − j +1 F = Z Σ k X i =0 C ik ( − i (cid:0) n − − k (cid:1) p k − i +1 F + Z Σ k X j =1 − j (cid:16) C jk j − C j − k ( k − j + 1) (cid:17) p k − j +1 F =( n − − k ) Z Σ k X i =0 C ik ( − i p k − i +1 =( n − − k ) Z Σ e N k F =( n − − k ) Z Σ e L k + ( n − − k ) Z (cid:16) e N k F − e L k (cid:17) . Substituting F = p k − p k into above, we get the desired result. (cid:3) In order to show the monotonicity of the functional Q defined in (1.21) under the inverse flow(2.7), we need to show the non-positivity of the last term in (4.1). That is p k − p k e N k − e L k ≤ . (4.2)When k = 1, (4.2) is just p p ( p − p ) − ( p − ≤ , which follows from the Newton-Maclaurin inequalities in Lemma 2.1. In fact, it is clear that p p ( p − p ) − ( p −
1) = ( p p p − p ) + (1 − p p ) . Hence the non-positivity follows, for both terms are non-positive, by Lemma 2.1. This was usedin [22]. When k ≥
2, the proof of (4.2) becomes more complicated. When k = 2, one needs toshow the non-positivity of(4.3) p p ( p − p + p ) − ( p − p + 1) = (cid:18) p p p − p (cid:19) + 2 (cid:18) p − p p (cid:19) + (cid:18) p p p − (cid:19) . By Lemma 2.1, the first two terms are non-positive, but the last term is non-negative. It wasshowed in [15] that (4.3) is non-positive if κ ∈ R n − satisfying(4.4) κ ∈ { κ = ( κ , κ , · · · , κ n − ) ∈ R n − | κ i ≥ } . We want to show that (4.2) is true for general k ≤ ( n − k = 2. Proposition 4.2.
For any κ satisfying (4.4), we have p k − p k e N k − e L k ≤ . (4.5) Equality holds if and only if one of the following two cases holdseither ( i ) κ i = κ j ∀ i, j, or ( ii ) ∃ i with κ i < κ j = 1 ∀ j = i. (4.6)We sketch the proof into several steps. Before the proof, we introduce the notation of P cyc tosimplify notations. Precisely, given n − κ , κ , · · · , κ n − ), we denote P cyc f ( κ , · · · , κ n − )the cyclic summation which takes over all different terms of the type f ( κ , · · · , κ n − ). For in-stance, X cyc κ = κ + κ + · · · + κ n − , X cyc κ κ = n − X i =1 (cid:16) κ i X j = i κ j (cid:17) , X cyc κ ( κ − κ ) = n − X i =1 (cid:18) κ i X ≤ j For any κ satisfying (4.4), we have (4.7) e N k − p e L k ≤ . Equality holds if and only if one of the following two cases holdseither ( i ) κ i = κ j ∀ i, j, or ( ii ) ∃ i with κ i > κ j = 1 ∀ j = i. Proof. It is crucial to observe that (4.7) is indeed equivalent to the following inequality: X ≤ i m ≤ n − ,i j = i l ( j = l ) κ i ( κ i κ i − κ i κ i − · · · ( κ i k − κ i k − − (cid:0) κ i k − κ i k +1 (cid:1) ≥ , (4.8)where the summation takes over all the (2 k + 1)-elements permutation of { , , · · · , n − } . Forthe convenience of the reader, we sketch the proof of (4.8) briefly. First, note that from (3.4)that ( p e L k − e N k ) = p k X i =0 C k − ik ( − k − i p i − k X i =0 C k − ik ( − k − i p i +1 = k X i =0 ( − k − i C ik ( p p i − p i +1 ) . Next we calculate each term p p i − p i +1 carefully. By using( n − C jn − = ( j + 1) C j +1 n − + ( n − C j − n − ) , YPERBOLIC ALEXANDROV-FENCHEL INEQUALITIES II 13 we have σ j σ = (cid:18) X cyc κ i κ i · · · κ i j (cid:19)(cid:18) X cyc κ i j +1 (cid:19) = ( j + 1) (cid:18) X cyc κ i κ i · · · κ i j κ i j +1 (cid:19) + X cyc κ i κ i · · · κ i j , and p p j − p j +1 = 1( n − C jn − (cid:18) X cyc κ i κ i · · · κ i j (cid:19)(cid:18) X cyc κ i j +1 (cid:19) − C j +1 n − X cyc κ i κ i · · · κ i j +1 = 1( n − C jn − C j +1 n − (cid:16) C j +1 n − (2 j + 1) X cyc κ i κ i · · · κ i j κ i j +1 + C j +1 n − X cyc κ i κ i · · · κ i j − ( n − C jn − X cyc κ i κ i · · · κ i j +1 (cid:17) = 1( n − C jn − C j +1 n − · C j +1 n − n − j X cyc κ i κ i · · · κ i j − ( κ i j − κ i j +1 ) = (2 j )!( n − j − n − · ( n − X cyc κ i κ i · · · κ i j − ( κ i j − κ i j +1 ) . In (4.8), the coefficient of κ κ · · · κ j − ( κ j − κ j +1 ) is2( − k − j C j − k − (2 j − C k − jn − j − [2( k − j )]! = ( − k − j k C jk (2 j )!( n − j − − k − j C jk (2 j )!( n − j − n − · ( n − · ( n − · ( n − k . Therefore we have0 ≤ X ≤ i m ≤ n − ,i j = i l ( j = l ) κ i ( κ i κ i − κ i κ i − · · · ( κ i k − κ i k − − (cid:0) κ i k − κ i k +1 (cid:1) = ( n − · ( n − k k X j =0 ( − k − j C jk ( p p j − p j +1 )= ( n − · ( n − k ( p e L k − e N k ) . This finishes the proof. (cid:3) In view of (4.8), we have the following remark which will be used later. Remark 4.4. For any κ = ( κ , · · · , κ n − ) satisfying < κ i ≤ , ( i = 1 , · · · , n − , then ( − k − (cid:0) e N k − p e L k (cid:1) ≤ . Lemma 4.5. For any κ satisfying (4.4), we have e N k ≥ , e L k ≥ . Proof. They are equivalent to the following inequalities respectively: X ≤ i m ≤ n − ,i j = i l ( j = l ) κ i ( κ i κ i − κ i κ i − · · · ( κ i k − κ i k − − κ i k κ i k +1 − ≥ , (4.9) X ≤ i m ≤ n − ,i j = i l ( j = l ) ( κ i κ i − κ i κ i − · · · ( κ i k − κ i k − − κ i k κ i k +1 − ≥ . (4.10)where the summation takes over all the (2 k + 1)-elements permutation of { , , · · · , n − } . Theproof to show the equivalence of (4.9),(4.10) is exactly the same as the one of (4.8). Hence weomit it here. (cid:3) Remark 4.6. For any κ = ( κ , · · · , κ n − ) satisfying < κ i ≤ , ( i = 1 , · · · , n − , then ( − k e N k ≥ , ( − k e L k ≥ . Making use of Lemma 4.3 and Remark 4.4, we can show the following result which is strongerthan Proposition 4.2. Lemma 4.7. For any κ satisfying (4.4), we have p k e N k − p k +1 e L k ≤ . Proof. According to the induction argument proved in [15] (see p.8), we only need to prove itfor n − k + 1. Let z i = κ i ≤ 1, andˆ p i = p i ( z , z , · · · , z k +1 ) . It is clear that ˆ p j = p k +1 − j p k +1 . (4.11)By Remark 4.4, we have( − k − k X i =0 C ik ( − i ˆ p k − i +1 − ( − k − ˆ p k X i =0 C ik ( − i ˆ p k − i ≤ , (4.12)which is equivalent to( − k − k X i =0 C ik ( − i p i p k +1 − ( − k − p k p k +1 k X i =0 C ik ( − i p i +1 p k +1 ≤ . (4.13)Thus we have k X i =0 C ik ( − k − i p i − p k p k +1 k X i =0 C ik ( − k − i p i +1 ≥ , (4.14)which implies p k p k +1 e N k − e L k ≤ . (cid:3) YPERBOLIC ALEXANDROV-FENCHEL INEQUALITIES II 15 Proof of Proposition 4.2. Then by the Newton-MacLaurin inequality p k − p k +1 ≤ p k , weobtain p k − p k e N k − e L k ≤ p k p k +1 e N k − e L k ≤ , which is exactly (4.5). Here we have used Lemma 4.5. (cid:3) Remark 4.8. Proposition 4.2 holds for κ ∈ R n − with κ i κ j ≥ for any i, j . This is equivalentto the condition that the sectional curvature of Σ is non-negative. Remark 4.9. From the proof of Proposition 4.2, it is easy to see that (4.5) has an inverseinequality for κ ∈ R n − with ≤ κ i ≤ . Now we have a monotonicity of Q (Σ t ) defined by (1.21) under the flow (2.7). Theorem 4.10. Functional Q is non-increasing under the flow (2.7) , provided that the initialsurface is horospherical convex.Proof. It follows from (3.1), (3.4) and Proposition 4.2 that(4.15) ddt Z Σ L k ≤ ( n − − k ) Z Σ L k . On the other hand, by (2.6) and (2.3), we also have(4.16) ddt | Σ t | = Z Σ t p k − p k ( n − p dµ ≥ ( n − | Σ t | . Combining (4.15) and (4.16) together, we complete the proof. (cid:3) Remark 4.11. From the above proof, one can check that to obtain a monotonicity of Q it isenough to choose F = p . Then from (4.1) and (4.7), it holds for all kddt Z e L k =( n − k − Z e L k + ( n − k − (cid:16) p e N k − e L k (cid:17) ≤ ( n − k − Z e L k . Proof of main Theorems Now we are ready to show our main theorems. Proof of Theorem 1.1. First recall the definition (1.21) of the functional Q , (1.13) is equivalentto(5.1) Q (Σ) ≥ C kn − (2 k )! ω kn − n − . Let Σ( t ) be a solution of flow (2.7) obtained by the work of Gerhardt [16]. This flow preservesthe horospherical convexity and non-increases for the functional Q . Hence, to show (5.1) weonly need to show(5.2) lim t →∞ Q (Σ t ) ≥ C kn − (2 k )! ω kn − n − . Since Σ is a horospherical convex hypersurface in ( H n , ¯ g ), it is written as graph of function r ( θ ), θ ∈ S n − . We denote X ( t ) as graphs r ( t, θ ) on S n − with the standard metric ˆ g . We set λ ( r ) = sinh( r ) and we have λ ′ ( r ) = cosh( r ). It is clear that( λ ′ ) = ( λ ) + 1 . We define ϕ ( θ ) = Φ( r ( θ )). Here Φ verifies Φ ′ = 1 λ . We define another function v = q |∇ ϕ | g . By [16], we have the following results. Lemma 5.1. λ = O ( e tn − ) , |∇ ϕ | + |∇ ϕ | = O ( e − tn − ) . From Lemma (5.1), we have the following expansions: λ ′ = λ (1 + 12 λ − ) + O ( e − tn − ) , (5.3)and 1 v = 1 − |∇ ϕ | g + O ( e − tn − ) . (5.4)We have also ∇ λ = λλ ′ ∇ ϕ. (5.5)The second fundamental form of Σ is written in an orthogonal basis (see [10] for example) h ij = λ ′ vλ (cid:18) δ ij − ϕ ij λ ′ + ϕ i ϕ l ϕ jl v λ ′ (cid:19) = δ ij + ( 12 λ − |∇ ϕ | ) δ ij − ϕ ij λ + O ( e − tn − ) , where the second equality follows from (5.3) and (5.4). We set T ij = ( 12 λ − |∇ ϕ | ) δ ij − ϕ ij λ , (5.6)then from the Gauss equations, we obtain R ij kl = − ( δ ik δ jl − δ il δ j k ) + ( h ik h j l − h il h j k )= δ ik T j l + T ik δ j l − T il δ jk − δ il T jk + O ( e − tn − ) . YPERBOLIC ALEXANDROV-FENCHEL INEQUALITIES II 17 It follows from (1.1) that L k = k δ i i ··· i k − i k j j ··· j k − j k R i i j j · · · R i k − i k j k − j k = 2 k δ i i ··· i k − i k j j ··· j k − j k T i j δ i j · · · T i k − j k − δ i k j k + O ( e − (2 k +2) tn − )= 2 k ( n − − k ) · · · ( n − k ) δ i i ··· i k − j j ··· j k − T i j T i j · · · T i k − j k − + O ( e − (2 k +2) tn − )= 2 k k !( n − − k ) · · · ( n − k ) σ k ( T ) + O ( e − (2 k +2) tn − ) . Here in the second equality we use the fact δ i i ··· i k − i k j j ··· j k − j k T i j δ i j = δ i i ··· i k − i k j j ··· j k − j k δ i j T i j = − δ i i ··· i k − i k j j ··· j k − j k T i j δ i j = − δ i i ··· i k − i k j j ··· j k − j k δ i j T i j , and in the third equality we use (2.1) and (3.3).Recall ϕ i = λ i /λλ ′ , then by (5.3) we have ϕ ij = λ ij λ − λ i λ j λ + O ( e − tn − ) . (5.7)By the definition of the Schouten tensor, A ˆ g = 1 n − (cid:18) Ric ˆ g − R ˆ g n − 2) ˆ g (cid:19) = 12 ˆ g. Its conformal transformation formula is well-known (see for example [30])(5.8) A λ ˆ g = − ∇ λλ + 2 ∇ λ ⊗ ∇ λλ − |∇ λ | λ ˆ g + A ˆ g = − ∇ λλ + 2 ∇ λ ⊗ ∇ λλ − |∇ λ | λ ˆ g + 12 ˆ g. Substituting (5.5) and (5.7) into (5.6), together with (5.8), we have T ij = (( λ ˆ g ) − A λ ˆ g ) ij + O ( e − tn − ) , which implies L k = 2 k k !( n − − k ) · · · ( n − k ) σ k ( A λ ˆ g ) + O ( e − (2 k +2) tn − ) . (5.9)As before, Σ( t ) is a horospherical convex hypersurface. As a consequence, Σ has the nonnegativesectional curvature so that T + O ( e − tn − ) is positive definite. We consider ˜ λ := λ − e − tn − andthe conformal metric ˜ λ ˆ g . We have˜ λ (˜ λ ˆ g ) − A ˜ λ ˆ g = 12 e − tn − I + 12 e − tn − (1 − e − tn − ) |∇ λ | λ I − e − tn − (1 − e − tn − )ˆ g − ∇ λ ⊗ ∇ λλ + λ (1 − e − tn − )( λ ˆ g ) − A λ ˆ g . Recall e − tn − I + λ (1 − e − tn − )( λ ˆ g ) − A λ ˆ g ∈ Γ + n − for the sufficiently large t and e − tn − (1 − e − tn − ) |∇ λ | λ I − e − tn − (1 − e − tn − )ˆ g − ∇ λ ⊗∇ λλ ∈ Γ + k for any k ≤ n − . Therefore, we infer ˜ λ ˆ g ∈ Γ + k for any k ≤ n − . The Sobolev inequality (2.9) for the σ k operator gives( vol (˜ λ ˆ g )) − n − − kn − Z S n − σ k ( A ˜ λ ˆ g ) dvol ˜ λ ˆ g ≥ ( n − · · · ( n − k )2 k k ! ω kn − n − . (5.10) On the other hand, we have( vol (˜ λ ˆ g )) − n − − kn − R S n − σ k ( A ˜ λ ˆ g ) dvol ˜ λ ˆ g = (1 + o (1))( vol ( λ ˆ g )) − n − − kn − R S n − σ k ( A λ ˆ g ) dvol λ ˆ g , (5.11)since λ − e − tn − = 1 + o (1) . As a consequence of (5.9),(5.10) and (5.11), we deducelim t → + ∞ ( vol (Σ( t ))) − n − − kn − Z Σ( t ) L k ≥ ( n − n − · · · ( n − k ) ω kn − n − . When (5.1) is an equality, then Q is constant along the flow. Then (4.16) is an equality, whichimplies that equality in the inequality p k − p k p ≥ , holds. Therefore, Σ is a geodesic sphere. (cid:3) Proof of Theorem 1.2. It follows from (3.1), (3.4) and Theorem 1.1 that when n − > k (5.12) Z Σ ˜ L k ≥ ω kn − n − ( | Σ | ) n − − kn − . Using the expression (3.5) of R Σ σ k in terms of R Σ ˜ L j we get the desired result Z Σ σ k ≥ C kn − ω n − ((cid:18) | Σ | ω n − (cid:19) k + (cid:18) | Σ | ω n − (cid:19) k n − − kn − ) k . By Theorem 1.1, equality holds if and only if Σ is a geodesic sphere.When n − k , since the hypersurface Σ is convex. we know that (1.13) is an equalitywhen n − k by the Gauss-Bonnet-Chern theorem, even for any hypersurface diffeomorphicto a sphere. Hence in this case, we also have all the above inequalities with equality which inturn implies by [22] or [15] that Σ is a geodesic sphere. (cid:3) Proof of Theorem 1.3. When n − > k , the proof follows directly from (5.12) and Lemma 3.3.When n − k , the proof follows by the same reason as in Theorem 1.2. (cid:3) From (1.17), it is easy to see that Theorem 1.3 implies Theorem 1.2, meanwhile Theorem 1.2may not directly imply Theorem 1.3, since there are negative coefficients in (3.8) above.6. Alexandrov-Fenchel inequality for odd k In this section, we show an Alexandrov-Fenchel inequality for σ , which follows from the resultof Cheng-Zhou [8] and Theorem 1.2 (or more precisely from [22]). YPERBOLIC ALEXANDROV-FENCHEL INEQUALITIES II 19 Theorem 6.1. Let n ≥ . Any horospherical convex hypersurface Σ ⊂ H n satisfies (6.1) Z Σ σ ≥ ( n − ω n − (cid:26)(cid:18) | Σ | ω n − (cid:19) + (cid:18) | Σ | ω n − (cid:19) n − n − (cid:27) . where ω n − is the area of the unit sphere S n − and | Σ | is the area of Σ . Equality holds if andonly if Σ is a geodesic sphere.Proof. Notice that the horospherical convex condition implies that the Ricci curvature of Σ isnon-negative. We observe first that by a direct computation (1.4) in [8] Z Σ | H − H | ≤ n − n − Z Σ | B − Hn − g | , is equivalent to Z Σ σ Z Σ σ ≤ n − n − (cid:0) Z Σ σ (cid:1) . (6.2)Then we use the optimal inequality for σ proved in [22] (see also Theorem 1.2), Z Σ σ ≥ ( n − n − (cid:18) ω n − n − | Σ | n − n − + | Σ | (cid:19) , (6.3)to obtain the desired inequality for σ , Z Σ σ ≥ ( n − ω n − (cid:26)(cid:18) | Σ | ω n − (cid:19) + (cid:18) | Σ | ω n − (cid:19) n − n − (cid:27) . When (6.1) is an equality, in turn, (6.3) is also a equality, then it follows from [22] that thehypersurface is a geodesic sphere. (cid:3) Motivated by Theorem 1.2 and (6.2), we would like propose the following Conjecture 6.2. Let n − ≥ k + 1 . Any horospherical convex hypersurface Σ ⊂ H n satisfies Z Σ σ k +1 ≥ C k +1 n − ω n − (cid:26)(cid:18) | Σ | ω n − (cid:19) k +1 + (cid:18) | Σ | ω n − (cid:19) k +1 ( n − k − n − (cid:27) k +12 . Equality holds if and only if Σ is a geodesic sphere. The conjecture follows from Theorem 1.2 and the following conjecture(6.4) (cid:0) C k +1 n − (cid:1) C k +2 n − C kn − Z Σ σ k +2 Z Σ σ k ≤ (cid:18) Z Σ σ k +1 (cid:19) . Acknowledgment. We would like to thank Wei Wang for his important help in the proof of themonotonicity of functional Q . References [1] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. reine angew.Math. (2007) 17–31.[2] T. 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Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, CNRS UMR 8050, D´epartement deMath´ematiques, Universit´e Paris Est-Cr´eteil Val de Marne,, 61 avenue du G´en´eral de Gaulle,94010 Cr´eteil Cedex, France E-mail address : [email protected] Albert-Ludwigs-Universit¨at Freiburg, Mathematisches Institut Eckerstr. 1 D-79104 Freiburg E-mail address : [email protected] School of Mathematical Sciences, University of Science and Technology of China Hefei 230026,P. R. China and Albert-Ludwigs-Universit¨at Freiburg, Mathematisches Institut Eckerstr. 1 D-79104 Freiburg E-mail address ::