On Rigidity of hypersurfaces with constant curvature functions in warped product manifolds
aa r X i v : . [ m a t h . DG ] D ec ON RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATUREFUNCTIONS IN WARPED PRODUCT MANIFOLDS
JIE WU AND CHAO XIA
Abstract.
In this paper, we first investigate several rigidity problems for hypersurfaces in thewarped product manifolds with constant linear combinations of higher order mean curvaturesas well as “weighted” mean curvatures, which extend the work [22, 5, 6] considering constantmean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the spaceforms with constant linear combinations of intrinsic Gauss-Bonnet curvatures L k . To achievethis, we develop some new kind of Newton-Maclaurin type inequalities on L k which may haveindependent interest. Introduction
The rigidity problem of hypersurfaces with constant curvature functions has attracted muchattention in the classical differential geometry. The most typical curvature functions are the extrinsic mean curvature and the intrinsic
Gauss (scalar) curvature. In 1899, Liebmann [21]showed two rigidity results that closed surfaces with constant Gauss curvature or convex closedsurfaces with constant mean curvature in R are spheres. Later, Süss [28] and Hsiung [17] provedthe rigidity for convex or star-shaped hypersurfaces in R n for all n . In later 1950s, the conditionof convexity or star-shapedness was eventually removed by Alexandrov in a series of papers [2].Namely, he proved that closed hypersurfaces with constant mean curvature embedded in theEuclidean space are spheres. This result is now often referred to as Alexandrov Theorem . Alsohis method, based on the maximum principle for elliptic equations, is totally different with allprevious ones and now referred to as
Alexandrov’s reflection method . The embeddedness conditionis necessary in view of the famous counterexamples provided by Hsiang-Teng-Yu [16] and Wente[29]. After the work of Alexandrov, lots of extensions appeared on such rigidity topic. Montieland Ros [26, 27, 24] proved results for hypersurfaces with constant higher order mean curvatures embedded in space forms, following the work of Reilly [25] who recovered Alexandrov Theorem byusing an integral technique. Simultaneously, Korevaar [19] proved the same results following themethod of Alexandrov. Later, Montiel [22] studied the same problem in more general ambientmanifolds, the warped product manifolds. His result was in fact Hsiung’s type since he addedthe condition of star-shapedness to the corresponding hypersurfaces. Quite recently, Brendle [5]removed this star-shapedness condition and hence proved Alexandrov Theorem for constant mean
Mathematics Subject Classification.
Primary 53C24, Secondary 52A20, 53C40.
Key words and phrases. constant mean curvature, rigidity, warped product manifold, Gauss-Bonnet curvature.The first author is partly supported by SFB/TR71 “Geometric partial differential equations” of DFG.The second author is supported by funding from the European Research Council under the European Union’sSeventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 267087. curvature hypersurfaces in general warped product manifolds, including the (Anti-)deSitter-Schwarzschild manifolds as a typical example. Thereafter, Brendle and Eichmair [6] extendedthe result to any compact star-shaped hypersurfaces with constant higher order mean curvature,where star-shapedness is needed again. For other generalizations, see for instance [1, 3, 4, 14,15, 23] and references therein.In this paper, we first investigate several related rigidity problems for hypersurfaces withconstant curvature functions embedded in the warped product manifolds.Let us start with the setting. Assume ( N n − ( K ) , g N ) is an ( n − -dimensional compactmanifold with constant sectional curvature K . Let ( M n , ¯ g ) be an n -dimensional ( n ≥ ) warpedproduct manifold M = [0 , ¯ r ) × λ N ( K ) (0 < ¯ r ≤ ∞ ) , equipped with a Riemannian metric ¯ g = dr + λ ( r ) g N , where λ : [0 , ¯ r ) → R is a smooth positive function satisfying the following conditions:(C1) λ ′ ( r ) > for all r ∈ (0 , ¯ r ) ;(C2) λ ′′ ( r ) λ ( r ) + K − λ ′ ( r ) λ ( r ) > for all r ∈ (0 , ¯ r ) ;(C3) λ ′′ ( r ) ≥ for all r ∈ (0 , ¯ r ) ;(C4) λ ′ (0) = 0 , λ ′′ (0) > ; λ ′′ ( r ) λ ( r ) − ( n − K − λ ′ ( r ) λ ( r ) is non-decreasing for r ∈ (0 , ¯ r ) .Condition (C2) is equivalent that Ricci curvature is smallest in the radial direction and thelatter part of (C4) is equivalent that scalar curvature is non-decreasing with respect to r (see(2.10) below). As shown in [5], the Schwarzschild, the (Anti-)deSitter-Schwarzschild and theReissner-Nordstrom manifolds satisfy (C1)-(C4).Before stating our results, let us give some notations and terminologies. For a hypersurface Σ in M , we denote by H k = H k ( λ ) the normalized k -th mean curvature of Σ , i.e., H k ( λ ) = 1 (cid:0) n − k (cid:1) σ k ( λ ) , (1.1)where λ = ( λ , · · · , λ n − ) are the principal curvatures of Σ and σ k is the k -th elementarysymmetric function. We say that Σ is k -convex if λ satisfies σ j ( λ ) ≥ for any ≤ j ≤ k . Σ is called star-shaped if h ∂∂r , ν i ≥ for the outward normal ν of Σ .Our first result is on hypersurfaces with constant curvature quotients in the warped productmanifolds. This kind of rigidity in the space forms can be obtained by using Alexandrov reflectionmethod, which was already referred by Korevaar [19]. Koh [18] gave another proof based on theMinkowski integral formula. Theorem 1.1.
Let ( M n , ¯ g ) be an n -dimensional ( n ≥ ) warped product manifold satisfying(C1) and (C2). Let ≤ l < k ≤ n − be two integers and Σ be a closed, star-shaped hypersurfacein ( M, ¯ g ) . If there exists some constant c such that H l is nowhere vanishing and H k H l ≡ c , then Σ is a slice N × { r } for some r ∈ (0 , ¯ r ) . Next, we study the rigidity problem for hypersurfaces with constant linear combinations ofmean curvatures in the warped product manifolds.
N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 3
Theorem 1.2.
Let ( M n , ¯ g ) be an n -dimensional ( n ≥ ) warped product manifold satisfying(C1) and (C2). Let ≤ l < k ≤ n − be two integers and Σ be a closed, k-convex star-shapedhypersurface in ( M n , ¯ g ) . If either of the following holds: (i) ≤ l < k ≤ n − and there are nonnegative constants { a i } l − i =1 and { b j } kj = l , at least oneof them not vanishing, such that l − X i =1 a i H i = k X j = l b j H j ; (ii) ≤ l < k ≤ n − and there are nonnegative constants { a i } l − i =0 and { b j } kj = l , at least oneof them not vanishing, such that l − X i =0 a i H i = k X j = l b j H j ; then Σ is a slice N × { r } for some r ∈ (0 , ¯ r ) . Theorems 1.1 and 1.2 will be proved by using the classical integral method due to Hsiung [17]and Reilly [25]. The main tools are Minkowski formulae as well as a family of Newton-Maclaurininequalities. Unlike in the space forms, the Newton tensor is generally not divergence-free in thewarped product manifolds. As observed in [6], the extra terms will have a good sign under thecondition (C2) and star-shapedness. However, to deal with our rigidity problems, one needs tokeep trail with these terms carefully rather than just throw them away. On the other hand, bythe generality of warped product manifolds, the classical Alexandrov’s reflection method [2] asin [19] seems to be difficult to deal with our problems.Next we will also study similar rigidity problems on some “weighted” higher mean curvaturesand their linear combinations. We denote the weight in the warped product manifolds by V ( r ) := λ ′ ( r ) . In [30], the first author discussed such rigidity result in H n . This kind of “weighted” meancurvature appears very naturally. Interestingly, the corresponding weighted Alexandrov-Fenchelinequalities relate to the quasi-local mass in H n and the Penrose inequalities for asymptoticallyhyperbolic graphs, see [7, 12] for instance. Our next result is regarding the above weightedrigidity results in the warped product manifolds. Theorem 1.3.
Let ( M n , ¯ g ) be an n -dimensional ( n ≥ ) warped product manifold satisfying(C1)-(C3). Let ≤ l < k ≤ n − be two integers and Σ n − be a closed star-shaped hypersurfacein ( M n , ¯ g ) . If one of the following case holds: (i) ( M, ¯ g ) satisfies (C4) and V H k is a constant for some k = 1 , · · · , n − ; (ii) ≤ l < k ≤ n − , Σ is k-convex and there are nonnegative constants { a i } l − i =1 and { b j } kj = l ,at least one of them not vanishing, such that l − X i =1 a i H i = k X j = l b j ( V H j ); JIE WU AND CHAO XIA (iii) ≤ l < k ≤ n − , Σ is k-convex and there are nonnegative constants { a i } l − i =0 and { b j } kj = l ,at least one of them not vanishing, such that l − X i =0 a i H i = k X j = l b j ( V H j ); then Σ is a slice N × { r } for some r ∈ (0 , ¯ r ) . Theorem 1.3 is proved in a similar way by taking the consideration of a new Minkowski typeformula, Proposition 2.3. We note that the presence of the weight makes Alexandrov’s reflectionmethod hard to apply even in the case of space forms, see [30].
Remark 1. (1)
Comparing with the results in [5, 6] , in the most cases we do not assume (C4). In fact,we mostly will not use the Heintze-Karcher type inequality derived in [5] , for which (C4)is essential. (2)
Theorem 1.2 contains the simplest case that H is constant. In view of Brendle’s resultin [5] , for this case, if one assumes further (C4) on M , the condition of -convexityand star-shapedness on hypersurfaces is actually superfluous. Similarly, the condition ofstar-shapedness is needless in Theorem 1.2 when we consider V H is a constant. (3) Theorem 1.2 also contain the case that higher order mean curvatures H k are constant.For this case, the k -convexity condition is superfluous since it is implied by the constancyof H k . (4) For similar rigidity problem in the space forms, the star-shapedness is not necessary. SeeTheorem 3.1 below.
The second part of this paper is about rigidity problems on some intrinsic curvature functionsof induced metric from that of the space forms. In fact, this is one of our motivations to studythe linear combinations of mean curvature functions. As mentioned at the beginning, Liebmann[21] showed closed surfaces with constant Gauss curvature in R are spheres. Apparently, inspace forms, one can see from the Gauss formula that surfaces with constant scalar (Gauss)curvature is equivalent to constant -nd mean curvature. Hence Liebmann’s result is equivalentto Ros’ [26]. On the other hand, there is a natural generalization of scalar curvature, calledGauss-Bonnet curvatures. The Pfaffian in Gauss-Bonnet-Chern formula is the highest orderGauss-Bonnet curvature. The general one appeared first in the paper of Lanczos [20] in 1938and has been intensively studied in the theory of Gauss-Bonnet gravity, which is a generalizationof Einstein gravity. Precisely, the Gauss-Bonnet curvatures are defined by(1.2) 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 , where δ i i ··· i k − i k j j ··· j k − j k is the generalized Kronecker delta defined in (2.4) below and R ijkl is theRiemannian curvature -tensor in local coordinates. It is easy to see that L is just the scalarcurvature R . When k = 2 , it is the second Gauss-Bonnet curvature L = k Rm k − k Ric k + R . N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 5
For general k it is the Euler integrand in the Gauss-Bonnet-Chern theorem if n = 2 k and istherefore called the dimensional continued Euler density in physics if k < n . Here n is thedimension of corresponding manifold. Using the Gauss-Bonnet curvatures one can define theGauss-Bonnet-Chern mass and guarantee its well-defineness in asymptotically flat manifolds aswell as asymptotically hyperbolic manifolds, see [9, 10, 12].In the Euclidean space R n , the intrinsic Gauss-Bonnet curvatures L k with the induced metricon the surfaces are the same with H k , up to some scaling constant. In the space forms rather than R n , L k can be expressed as some linear combination of H k (see Lemma 4.1 below). Explicitly,for the unit sphere S n , L k = (cid:18) n − k (cid:19) (2 k )! k X i =0 (cid:18) ki (cid:19) H k − i . Notice here all the coefficients are positive. Therefore as a direct consequence of Theorem 3.1(ii), we have the following
Corollary 1.4.
Let ≤ k ≤ n − be an integer and Σ be a closed k -convex hypersurfaceembedded in the hemisphere S n + . If the k -th Gauss-Bonnet curvature L k is constant, then Σ is acentered geodesic hypersphere. Unlike in S n , the intrinsic Gauss-Bonnet curvature L k in H n is a linear combination of H k with sign-changed coefficients. Precisely, L k = (cid:18) n − k (cid:19) (2 k )! k X i =0 (cid:18) ki (cid:19) ( − i H k − i . Hence we cannot apply Theorem 3.1 (ii) directly to conclude the rigidity. Moreover, we couldprove the general rigidity result of hypersurfaces in terms of the constant linear combinationsof L k . This rigidity of combination form is not direct which evolves the development of somenew kind Newton-Maclaurin type inequalities on L k rather than H k (see Proposition 4.2 andPropositon 4.4 below ), for horoconvex hypersurfaces. Here a hypersurface in H n is horosphericalconvex if all its principal curvatures are larger than or equal to . The horospherical convexityis a natural geometric concept, which is equivalent to the geometric convexity in Riemannianmanifolds. Theorem 1.5.
Let ≤ l < k ≤ n − be two integers and Σ be a closed horospherical convexhypersurface in the hyperbolic space H n . If there are nonnegative constants { a i } l − i =0 and { b j } kj = l ,at least one of them not vanishing, such that l − X i =0 a i L i = k X j = l b j L j , then Σ is a centered geodesic hypersphere. In particular, if L k is constant, then Σ is a centeredgeodesic hypersphere. For S n + , we can also establish similar Newton-Maclaurin type inequalities for k -convex hyper-surfaces, which enables us to prove rigidity in the hemisphere S n + for a general linear combinationof curvatures, as in H n . JIE WU AND CHAO XIA
Theorem 1.6.
Let ≤ l < k ≤ n − be two integers and Σ be a closed k -convex hypersurfaceembedded in the hemisphere S n + . If there are nonnegative constants { a i } l − i =0 and { b j } kj = l , at leastone of them not vanishing, such that l − X i =0 a i L i = k X j = l b j L j , then Σ is a centered geodesic hypersphere. Note that Theorem 1.6 is an extension of Corollary 1.4. However, it does not follow directlyfrom Theorem 3.1 below.The paper is organized as follows. In Section 2, we provide several preliminary results includingthe most important tool of this paper, Minkowski type formulae. Section 3 is devoted to proveour main theorems of the first part, Theorems 1.1-1.3. In Section 4, we focus on the rigidityproblem on the intrinsic Gauss-Bonnet curvatures and show Theorems 1.5 and 1.6.2.
Preliminaries
In this section, let us first recall some basic definitions and properties of higher order meancurvature.Let σ k be the k -th elementary symmetry function σ k : R n − → R defined by σ k (Λ) = X i < ···
Here the generalized Kronecker delta is defined by(2.4) 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 . We use the convention that T − = 0 . The k -th positive Garding cone Γ + k is defined by(2.5) Γ + k = { Λ ∈ R n − | σ j (Λ) > , ∀ j ≤ k } . And its closure is denoted by Γ + k . A symmetric matrix B is said to belong to Γ + k if λ ( B ) ∈ Γ + k .Let(2.6) H k = σ k (cid:0) n − k (cid:1) , be the normalized k -th elementary symmetry function. As a convention, we take H = 1 , H − =0 . The following Newton-Maclaurin inequalities are well known. For a proof, we refer to a surveyof Guan [13]. Lemma 2.1.
For ≤ l < k ≤ n − and Λ ∈ Γ + k , the following inequalities hold: H k − H l ≥ H k H l − . (2.7)(2.8) H l ≥ H lk k . Moreover, equality holds in (2.7) or (2.8) at Λ if and only if Λ = c (1 , , · · · , for some c ∈ R . Next, we collect some well-known results for the warped product manifold ( M = [0 , ¯ r ) × λ N ( K ) , ¯ g ) .We denote by ¯ ∇ and ∇ the covariant derivatives on M n and the surface Σ respectively. As in[5, 6], we define a smooth function V : M → R and a vector field X on M by V ( r ) = λ ′ ( r ) and X = λ ( r ) ∂∂r . Note that X is a conformal vector field satisfying ¯ ∇ X = V ¯ g. (2.9)Condition (C1) implies that V is a positive function on (0 , ¯ r ) × N ( K ) . One can verify that everyslice { r } × N ( K ) , r ∈ (0 , ¯ r ) , has constant principal curvatures λ ′ ( r ) λ ( r ) > . The Ricci curvature of ( M, ¯ g ) is given by Ric = − (cid:18) λ ′′ ( r ) λ ( r ) − ( n − K − λ ′ ( r ) λ ( r ) (cid:19) ¯ g − ( n − (cid:18) λ ′′ ( r ) λ ( r ) + K − λ ′ ( r ) λ ( r ) (cid:19) dr ⊗ dr. (2.10)Let { e i } n − i =1 and ν be an orthonormal basis and the outward normal of Σ respectively. Denoteby h ij the second fundamental form of Σ with this basis and λ = ( λ , · · · , λ n − ) the principal JIE WU AND CHAO XIA curvatures of Σ . The star-shapedness of Σ means h ∂∂r , ν i ≥ . (2.11)We need the following Minkowski type formula in the product manifolds, which is included inthe proof of [6], Proposition 8 and Proposition 9. For completeness, we involve a proof here. Proposition 2.2.
Let Σ n − be a closed hypersurface isometric immersed in the product manifold ( M, ¯ g ) . Then (i) we have Z Σ h X, ν i H k dµ = Z Σ V H k − dµ + k − (cid:0) n − k − (cid:1) Z Σ n − X i,j =1 A ij ( T k − ) ij dµ, ∀ ≤ k ≤ n − , (2.12) where A ij := − n − n − h X, e i i Ric ( e j , ν ) . (2.13)(ii) If Σ is star-shaped and ( M, ¯ g ) satisfies (C2), then we have A jj ≥ , ∀ ≤ j ≤ n − . (2.14) Proof. (i) It follows from the Gauss-Weingarten formula and (2.9) that ∇ i X j = ¯ ∇ i X j − h X, ν i h ij = V ¯ g ij − h X, ν i h ij . (2.15)Multiplying (2.15) by the k -th Newton transform tensor ( T k − ) ij and summing over i, j , weobtain n − X i,j =1 ∇ i ( X j ( T k − ) ij ) = n − X j =1 X j n − X i =1 ∇ i ( T k − ) ij + V n − X i,j =1 ( T k − ) ij ¯ g ij − h X, ν i n − X i,j =1 ( T k − ) ij h ij = n − X j =1 X j n − X i =1 ∇ i ( T k − ) ij + V ( n − k ) σ k − − kσ k h X, ν i , (2.16)where (2.2) and (2.3) are used to get (2.16).By the definition of ( T k − ) ij , we know that n − X i =1 ∇ i ( T k − ) ij = n − X i =1 n − X i , ··· ,i k − =1 ,j , ··· ,j k − =1 k − δ i ··· i k − ij ··· j k − j ∇ i h i j · · · h i k − j k − = n − X i =1 n − X i , ··· ,i k − =1 ,j , ··· ,j k − =1
12 1( k − δ i ··· i k − ij ··· j k − j ( ∇ i h i j − ∇ i h ij ) h i j · · · h i k − j k − . (2.17) N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 9 As N ( K ) is of constant sectional curvature, it is easy to see that M is locally conformally flat.Using Codazzi equation and the local conformal flatness of M , we have ∇ i h i j − ∇ i h ij = Riem ( e i , e i , e j , ν )= − n − Ric ( e i , ν ) δ i j − Ric ( e i , ν ) δ ij ) . (2.18)Substituting (2.17) into (2.18), we deduce that n − X i =1 ∇ i ( T k − ) ij = − n − n − X i =1 Ric ( e i , ν ) n − X i , ··· ,i k − =1 ,j , ··· ,j k − =1 ,j = j,j , ··· ,j k − k − δ j i ··· i k − ij j ··· j k − j h i j · · · h i k − j k − = − n − kn − n − X i =1 Ric ( e i , ν ) n − X i , ··· ,i k − =1 ,j , ··· ,j k − =1 k − δ i ··· i k − ij ··· j k − j h i j · · · h i k − j k − = − n − kn − n − X i =1 Ric ( e i , ν )( T k − ) ij . (2.19)Now by taking integration of (2.16) over Σ together with (2.19) and taking (2.6) into account,we arrive at (2.12).(ii) We know from (2.10) that Ric ( e j , ν ) = − ( n − (cid:18) λ ′′ ( r ) λ ( r ) + K − λ ′ ( r ) λ ( r ) (cid:19) λ ( r ) h X, e j ih X, ν i , which implies A jj = − n − n − h X, e j i Ric ( e j , ν )= 1( n − (cid:18) λ ′′ ( r ) λ ( r ) + K − λ ′ ( r ) λ ( r ) (cid:19) λ ( r ) h X, e j i h X, ν i . By using the star-shapedness (2.11) of Σ and the assumption (C2) on λ ( r ) , we conclude A jj ≥ for any j = 1 , · · · , n − . (cid:3) For later purpose to prove the rigidity result on weighted curvature functions, we need toextend the above proposition to the following type.
Proposition 2.3.
Let Σ be a hypersurface isometric immersed in the product manifold ( M n , ¯ g ) ,we have (2.20) Z Σ h X, ν i V H k dµ = Z Σ V H k − dµ + k − (cid:0) n − k − (cid:1) Z Σ n − X i,j =1 V A ij ( T k − ) ij dµ + 1 k (cid:0) n − k (cid:1) Z Σ ( T k − ) ij X i ∇ j V dµ.
Moreover, if Σ is k-convex and ( M, ¯ g ) satisfies condition (C3), then we have (2.21) Z Σ h X, ν i V H k dµ ≥ Z Σ V H k − dµ + k − (cid:0) n − k − (cid:1) Z Σ n − X i,j =1 V A ij ( T k − ) ij dµ. Equality holds if and only if Σ is totally umbilical in ( M n , ¯ g ) .Proof. Combining (2.16) and (2.19) together, we arrive at k (cid:0) n − k (cid:1) ∇ i (cid:0) ( T k − ) ij X j (cid:1) = −h X, ν i H k + V H k − + k − (cid:0) n − k − (cid:1) n − X i,j =1 A ij ( T k − ) ij , (2.22)where A ij is defined in (2.13). Multiplying above equation by the function V and integrating byparts, one obtains the desired result (2.20). Noting that X i = λ ( r ) ∇ i r, ∇ j V = λ ′′ ( r ) ∇ j r, we have ( T k − ) ij X i ∇ j V = λ ( r ) λ ′′ ( r )( T k − ) ij ∇ i r ∇ j r. (2.23)Under the assumption that Σ is k -convex, the ( k − -th Newton tensor T k − is positively definite(see e.g. Guan [13]), hence ( T k − ) ij ∇ i r ∇ j r ≥ . Together with assumption (C3) λ ′′ ( r ) ≥ , (2.21) holds. When the equality holds, we have ∇ r = 0 which implies that Σ is umbilical in ( M n , ¯ g ) . (cid:3) Finally, we need a Heintze-Karcher-type inequality due to Ros [27] and Brendle [5].
Proposition 2.4 (Brendle) . Let ( M n = [0 , ¯ r ) × N ( K ) , ¯ g = dr + λ ( r ) g N ) be a warped productspace satisfying (C1),(C2),(C4), or one of the space forms R n , S n + , H n . Let Σ be a compacthypersurface embedded in ( M n , ¯ g ) with positive mean curvature H , then Z Σ h X, ν i dµ ≤ Z Σ VH dµ. Moreover, equality holds if and only if Σ is totally umbilical. Rigidity for curvature quotients and combinations
In this section, we are ready to prove our main theorems. We start with the one on curvaturequotients. This will be proved by making use of Lemma 2.1 and Proposition 2.2.
Proof of Theorem 1.1:
We first claim that λ ∈ Γ + k . In fact, condition (C1) implies that Σ hasat least one elliptic point where all the principal curvatures are positive. This can be shownby a standard argument using maximum principle. Hence the constant c should be positive.Moreover, since H l is nowhere vanishing on Σ , it must be positive. In turn, H k = cH l is positive.From the result of Gårding [8], we know that H j > everywhere on Σ for ≤ j ≤ k . N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 11
For ≤ l < k ≤ n − , Proposition 2.2 gives the following two formulae: Z Σ h X, ν i H k dµ = Z Σ V H k − dµ + k − (cid:0) n − k − (cid:1) Z Σ n − X i,j =1 A ij ( T k − ) ij dµ, (3.1) Z Σ h X, ν i H l dµ = Z Σ V H l − dµ + l − (cid:0) n − l − (cid:1) Z Σ n − X i,j =1 A ij ( T l − ) ij dµ. (3.2)Since H k = cH l , we deduce from (3.1),(3.2) together with (2.7), (2.14) that Z Σ h X, ν i ( H k − cH l ) dµ = Z Σ V ( H k − − cH l − ) dµ + Z Σ n − X i,j =1 A ij k − (cid:0) n − k − (cid:1) ( T k − ) ij − c l − (cid:0) n − l − (cid:1) ( T l − ) ij ! dµ. (3.3)Without loss of generality, one may assume that the second fundamental form h ij is diagonal atthe point under computation. At this point, we have n − X i,j =1 A ij k − (cid:0) n − k − (cid:1) ( T k − ) ij − c l − (cid:0) n − l − (cid:1) ( T l − ) ij ! = n − X j =1 A jj (( k − H k − (Λ j ) − c ( l − H l − (Λ j )) . (3.4)We know from the Newton-Maclaurin inequality (2.7) that H k − H l − ≥ H k H l = c. (3.5)On the other hand, note the simple fact σ k = λ j σ k − (Λ j ) + σ k (Λ j ) , which is equivalent to H k = kn − λ j H k − (Λ j ) + n − − kn − H k (Λ j ) . (3.6)Applying (3.6), for any j = 1 , · · · , n − , we find ( k − H k − (Λ j ) H l − − ( l − H k − H l − (Λ j )= ( k − n − l ) n − H k − (Λ j ) H l − (Λ j ) − ( l − n − k ) n − H l − (Λ j ) H k − (Λ j )= ( k − l ) H k − (Λ j ) H l − (Λ j )+ ( l − n − k ) n − H k − (Λ j ) H l − (Λ j ) − H l − (Λ j ) H k − (Λ j )) > . (3.7) Therefore, by (2.14), (3.4), (3.5) and (3.7), the integrand in the right hand side of (3.3) is non-negative. It follows that the equality holds in (3.5), which implies that Σ is totally umbilical.Moreover, thanks to (3.7), we have A jj ≡ , ∀ ≤ j ≤ n − . (3.8)Together with condition (C2), (3.8) implies that the normal ν is parallel or pendicular to ∂∂r everywhere on Σ . However, there is at least one point on Σ where ν is parallel to ∂∂r . Therefore, ν is parallel to ∂∂r for all points in Σ , which means that Σ is a slice { r } × N ( K ) . We completethe proof. (cid:3) Next we show the rigidity result for constant linear combinations of mean curvatures in thewarped product manifolds. This argument basically follows from the above one except that oneneeds pay more attention to the use of the Newton-Maclaurin inequality at the first step.
Proof of Theorem 1.2: (i) By the existence of an elliptic point and non-vanishing of at least one coefficient, we know P l − i =1 a i H i > . Since Σ is k-convex, we recall from (2.7) that H i H j − ≥ H i − H j , ≤ i < j ≤ k, (3.9)where all equalities hold if and only if Σ is umbilical. Multiplying (3.9) by a i and b j and summingover i and j , we get l − X i =1 a i H i k X j = l b j H j − ≥ l − X i =1 a i H i − k X j = l b j H j . (3.10)By using the assumption l − X i =1 a i H i = k X j = l b j H j > , ≤ l < k ≤ n − , we obtain from (3.10) that k X j = l b j H j − ≥ l − X i =1 a i H i − . (3.11)On the other hand, (2.7) and (3.7) give ( j − H j − (Λ p ) H i > ( i − H j H i − (Λ p ) , ∀ ≤ i < j ≤ k, ≤ p ≤ n − . (3.12)Multiplying (3.12) by a i and b j and summing over i and j , we have k X j = l ( j − b j H j − (Λ p ) l − X i =1 a i H i > k X j = l b j H j l − X i =1 ( i − a i H i − (Λ p ) . N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 13
Hence k X j = l ( j − b j H j − (Λ p ) > l − X i =1 ( i − a i H i − (Λ p ) , ∀ ≤ p ≤ n − . (3.13)As in the proof of Theorem 1.1, (3.13) implies the matrix k X j = l ( j − (cid:0) n − j − (cid:1) b j ( T j − ) pq − l − X i =1 ( i − (cid:0) n − i − (cid:1) a i ( T i − ) pq n − p,q =1 is positive definite.(3.14)We finally infer from (3.1), (3.2) that Z Σ ( k X j = l b j H j − l − X i =1 a i H i ) h X, ν i dµ = Z Σ ( k X j = l b j H j − − l − X i =1 a i H i − ) V dµ + Z Σ n − X p,q =1 A pq k X j = l ( j − (cid:0) n − j − (cid:1) b j ( T j − ) pq − l − X i =1 ( i − (cid:0) n − i − (cid:1) a i ( T i − ) pq dµ ≥ . (3.15)Here the last inequality follows from (2.14), (3.4) (3.11) and (3.14).(ii) The proof is essentially the same as above. One only needs to notice the slight differenceregarding the value of indices. Proceeding as above, we have k X j = l b j H j +1 ≤ l − X i =0 a i H i +1 , (3.16)and k X j = l jb j H j − (Λ p ) > l − X i =0 ia i H i − (Λ p ) , ∀ ≤ p ≤ n − . (3.17)Applying (2.12) again, Z Σ ( l − X i =0 a i H i − k X j = l b j H j ) V dµ = Z Σ ( l − X i =0 a i H i +1 − k X j = l b j H j +1 ) h X, ν i dµ + Z Σ n − X p,q =1 A pq k X j = l jb j (cid:0) n − j − (cid:1) ( T j − ) pq − l − X i =0 ia i (cid:0) n − i − (cid:1) ( T i − ) pq dµ ≥ . (3.18)Here the last inequality follows from (2.14), (3.4), (3.16) and (3.17).We finish the proof by examining the equality in both cases as in the proof in Theorem 1.1. (cid:3) As remarked in the introduction, for the same rigidity problem in the space forms, the star-shapedness is not necessary. That is, we have the following theorem.
Theorem 3.1.
Let ≤ k ≤ n − be an integer and Σ n − be a closed, k-convex hypersurface in R n ( S n + , H n , resp. ) . If either of the following case holds: (i) ≤ l < k ≤ n − and there are nonnegative constants { a i } l − i =1 and { b j } kj = l , at least oneof them not vanishing, such that l − X i =1 a i H i = k X j = l b j H j ; (ii) there are nonnegative constants a and { b j } kj =1 , at least one of them not vanishing, suchthat a = k X j =1 b j H j ; then Σ is a geodesic hypersphere. For the proof of Theorem 3.1, we still apply the integral technique following [25]. We remarkthat it could be also obtained by using the classical Alexandrov’s reflection method as in [19].For the space forms R n ( S n + , H n resp.), the conformal vector field X = r ∂∂r ( sin r ∂∂r , sinh r ∂∂r resp.) and V = 1 ( cos r , cosh r resp.). It follows from the Codazzi equation that the Newtontensor T k is divergence-free with the induced metric on Σ , i.e., ∇ i T ijk = 0 . Thus (2.16) implies(3.19) ∇ j ( T ijk − X i ) = − k h X, ν i σ k + ( n − k ) V σ k − . Integrating above equation and noting (2.6), we have the Minkowski formula in the space forms(3.20) Z Σ h X, ν i H k dµ = Z Σ V H k − dµ. Proof of Theorem 3.1: (i)It follows from (3.20) and (2.7) that Z Σ h X, ν i ( l − X i =1 a i H i − k X j = l b j H j ) dµ = Z Σ V ( l − X i =1 a i H i − − k X j = l b j H j − ) dµ ≤ . (3.21)The last inequality follows from (3.11), where equality holds if and only if Σ is a geodesichypersphere.(ii) From the existence of an elliptic point and non-vanishing of at least one coefficient, wehave P kj =1 b j H j > . Since Σ is k -convex, k X j =1 b j H j ≥ k X j =1 b j H j > . N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 15
Hence H cannot vanish at any points, which implies that H > . Making use of (3.20) and(2.7), we derive a Z Σ h X, ν i dµ = Z Σ h X, ν i k X j =1 b j H j dµ = Z Σ V k X j =1 b j H j − H H dµ ≥ Z Σ V k X j =1 b j H j H dµ = a Z Σ VH dµ ≥ a Z Σ h X, ν i dµ, where in the last inequality we used Proposition 2.4. Therefore, the equality in both case yieldsthat Σ is a geodesic hypersphere. (cid:3) Using a similar argument and taking Propositions 2.3 and 2.4 into account, we now prove therigidity for the weighted curvature functions.
Proof of Theorem 1.3: (i) First the existence of an elliptic point implies that H k is positiveeverywhere on Σ . Then we know that H j > and H j (Λ p ) ≥ , ∀ ≤ p ≤ n − for ≤ j ≤ k .Thus (2.21) implies(3.22) Z Σ h X, ν i V H k dµ ≥ Z Σ V H k − dµ. Noticing from (2.8) that H k − ≥ H k − k k , we compute V H k Z Σ h X, ν i dµ = Z Σ h X, ν i V H k dµ ≥ Z Σ V H k − dµ ≥ Z Σ V H k − k k dµ = ( V H k ) k − k Z Σ V k dµ, which yields(3.23) Z Σ h X, ν i dµ ≥ ( V H k ) − k Z Σ V k dµ, and equality holds if and only Σ is a geodesic sphere.On the other hand, by Proposition 2.4 and (2.8) we derive that(3.24) Z Σ h X, ν i dµ ≤ Z Σ VH dµ ≤ Z Σ VH k k dµ = ( V H k ) − k Z Σ V k dµ. Finally combining (3.23) and (3.24) together, we complete the proof. (ii) As in the proof of Theorem 1.2, one can obtain the following two inequalities: k X j = l b j ( V H j − ) ≥ l − X i =1 a i H i − , (3.25)and k X j = l ( j − b j V H j − (Λ p ) > l − X i =1 ( i − a i H i − (Λ p ) , ∀ ≤ p ≤ n − . (3.26)For ≤ l < k ≤ n − , it follows from Proposition 2.2 and Proposition 2.3 that Z Σ h X, ν i V H k dµ ≥ Z Σ V H k − dµ + k − (cid:0) n − k − (cid:1) Z Σ n − X p,q =1 V A pq ( T k − ) pq dµ, (3.27) Z Σ h X, ν i H l dµ = Z Σ V H l − dµ + l − (cid:0) n − l − (cid:1) Z Σ n − X p,q =1 V A pq ( T l − ) pq dµ. (3.28)We then derive from above that Z Σ ( k X j = l b j V H j − l − X i =1 a i H i ) h X, ν i dµ = Z Σ V ( k X j = l b j V H j − − l − X i =1 a i H i − ) dµ (3.29) + Z Σ n − X p,q =1 A pq k X j = l ( j − (cid:0) n − j − (cid:1) b j V ( T j − ) pq − l − X i =1 ( i − (cid:0) n − i − (cid:1) a i ( T i − ) pq dµ ≥ . (3.30)Here the last inequality follows from (2.14), (3.4), (3.25) and (3.26). We finish the proof byexamining the equality case as before.(iii) The proof is similar with above with some necessary adaption as the one did in the proofof Theorem 1.2 (ii). (cid:3) rigidity for L k curvatures and their combinations Unlike the mean curvatures H k , the Gauss-Bonnet curvatures L k , and hence R Σ L k dµ areintrinsic geometric quantities, which depend only on the induced metric on Σ and are independentof the embeddings of Σ . The functionals R Σ L k are new geometric quantities for the study of theintegral geometry in the space forms.We first infer a relation between L k and H k . Lemma 4.1.
For a hypersurface (Σ , g ) in the space forms H n ( R n , S n , resp.) with constantcurvature ǫ = − , , resp. ) , its Gauss-Bonnet curvature L k with respect to g can be expressed N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 17 by higher order mean curvatures L k = (cid:18) n − k (cid:19) (2 k )! k X i =0 (cid:18) ki (cid:19) ǫ i H k − i . (4.1) Proof.
First by the Gauss formula R ij kl = ( h ik h j l − h il h j k ) + ǫ ( δ ik δ j l − δ il δ j k ) , where h ij := g ik h kj and h is the second fundamental form. Then substituting the Gauss formulaabove into (1.2) and noting (2.2), a straightforward calculation leads to, 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 + ǫδ i j δ i j ) · · · ( h i k − j k − h i k j k + ǫδ i k − j k − δ i k j k )= k X i =0 (cid:18) ki (cid:19) ǫ i ( n − k )( n − k + 1) · · · ( n − − k + 2 i ) (cid:0) (2 k − i )! σ k − i (cid:1) = (cid:18) n − k (cid:19) (2 k )! k X i =0 (cid:18) n − i (cid:19) ǫ i H k − i . Here in the second equality we used the symmetry of generalized Kronecker delta and in thethird equality we used (2.2) 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 . (4.2) (cid:3) Motivated by the expression (4.1), we introduce the following notations,(4.3) e L k := L k (cid:0) n − k (cid:1) (2 k !) , e N k := N k (cid:0) n − k (cid:1) (2 k !) , where N k := (cid:18) n − k (cid:19) (2 k )! k X i =0 (cid:18) ki (cid:19) ǫ i H k − i +1 . Since for the sphere S n , L k can be expressed as linear combinations of H k with nonnegativecoefficients in the formula (4.1), thus rigidity for L k is an immediate consequence of Theorem3.1. Proof of Corollary 1.4:
In the hemisphere S n + , there exists an elliptic point. Thus L k = const. isequivalent to k X i =1 (cid:18) ki (cid:19) H i = a , for some positive a . Hence the conclusion follows from Theorem 3.1. (cid:3) However, the hyperbolic case is not that easy. We will apply a new kind of Newton-Maclaurintype inequality to the hyperbolic case. It is clear that in hyperbolic space(4.4) e L k = k X i =0 (cid:18) ki (cid:19) ( − k − i H i , e N k = k X i =0 (cid:18) ki (cid:19) ( − k − i H i +1 . Due to the sign-changed coefficients of L k in terms of H k , it seems to be difficult to applyNewton-Maclaurin inequalities directly. Fortunately, under the condition of horoconvexity, wehave the following refined Newton-Maclaurin inequalities [11]. Proposition 4.2.
For any κ satisfying κ ∈ { κ = ( κ , κ , · · · , κ n − ) ∈ R n − | κ i ≥ } , we have (4.5) e N k − H 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 ) k ≥ , ∃ i with κ i > κ j = 1 ∀ j = i. Proof.
This proposition is proved in [11]. The key point is to observe that (4.5) is equivalent tothe 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) ≥ , (4.6)where the summation takes over all the (2 k + 1) -elements permutation of { , , · · · , n − } . Werefer the readers to [11] for more details. (cid:3) With all above preparing work, we are ready to prove a special case of Theorem 1.5 first.
Theorem 4.3.
Let ≤ k ≤ n − be an integer and Σ n − be a closed horospherical convexhypersurface in the hyperbolic space H n . If L k with the induced metric on Σ is constant, then Σ is a centered geodesic hypersphere.Proof. Since L = R = ( n − n − H − , it suffices to discuss the remaining case k ≥ . Observe that (3.20) implies Z Σ V e L k dµ = Z Σ h X, ν i e N k dµ. (4.7)The definition of e L k , e N k gives e L = 1 , e N = H . Thus using (3.20) again, we have Z Σ V e L dµ = Z Σ h X, ν i e N dµ. (4.8)By (4.1), we know that e L k is also constant. Combining (4.7) and (4.8) together, we have Z Σ h X, ν i ( e N k − e N e L k ) = 0 . N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 19
On the other hand, (4.5) yields e N k − e N e L k ≤ . This forces e N k − e N e L k = 0 . everywhere in Σ . By Proposition 4.2, there are two cases that equality holds. However, we assertthat the second case will not happen. In fact, in case ( ii ) we have from (4.10) below that e L k ≡ , ∀ k ≥ . However, in H n , there exists a horo-elliptic point, where all principal curvatures are strictlylarger than (this follows from the fact λ ′ ( r ) /λ ( r ) > ). Hence it follows again from (4.10)below that at this point e L k > . We get a contradiction. Therefore we conclude that Σ is ageodesic sphere. (cid:3) To prove the rigidity result regarding the general linear combination of L k , Proposition 4.2 isnot enough. We need to develop the following more general Newton-Maclaurin type inequalitieswhich may have independent interest. Proposition 4.4.
For any κ satisfying κ ∈ { κ = ( κ , κ , · · · , κ n − ) ∈ R n − | κ i ≥ } , we have (4.9) e N k − e L k ≥ e N k 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 ) k ≥ , ∃ i with κ i > κ j = 1 ∀ j = i. Proof.
Set κ i = 1 + ˆ κ i , then ˆ κ i ≥ for any i ∈ { , · · · , n − } . Define ˆ H i := H i (ˆ κ , ˆ κ , · · · , ˆ κ n − ) . Then H k = k X i =0 (cid:18) ki (cid:19) ˆ H i . thus e L k = k X i =0 i (cid:18) ki (cid:19) ˆ H k − i , e N k = k X i =0 i (cid:18) ki (cid:19) ˆ H k +1 − i . (4.10) Observing that e L k and e N k can be splitted into two terms, e L k = k − X i =0 i (cid:18) k − i (cid:19) ˆ H k − i + 2 k − X i =0 i (cid:18) k − i (cid:19) ˆ H k − − i , e N k = k − X i =0 i (cid:18) k − i (cid:19) ˆ H k +1 − i + 2 k − X i =0 i (cid:18) k − i (cid:19) ˆ H k − i , we introduce the notation X s,t =: t X i =0 i (cid:18) ti (cid:19) ˆ H s + t − i . It is clear that e L k = X k +1 ,k − + 2 X k,k − , e L k − = X k − ,k − , e N k = X k +2 ,k − + 2 X k +1 ,k − , e N k − = X k,k − . Hence the desired result (4.9) is equivalent to(4.11) ( X k +1 ,k − + 2 X k,k − ) X k,k − ≥ ( X k +2 ,k − + 2 X k +1 ,k − ) X k − ,k − . We claim that this is true. In fact, we can show the more general result as stated in the followinglemma:
Lemma 4.5.
For any s ≥ and t ≥ , (4.12) X s,t ≥ X s +1 ,t X s − ,t . Proof.
We use the induction argument for t to prove this lemma. When t = 0 , (4.12) holds forany s ≥ by the standard Newton-MacLaurin identity (2.7). Assume (4.12) holds for t , we needto prove that (4.12) holds for t + 1 . Observe the relation that(4.13) X s,t +1 = X s +1 ,t + 2 X s,t . Using the assumption that (4.12) holding for any s ≥ and fixed t , we derive X s,t +1 − X s +1 ,t +1 X s − ,t +1 = (cid:0) X s +1 ,t − X s +2 ,t X s,t (cid:1) + 2 ( X s +1 ,t X s,t − X s +2 ,t X s − ,t ) + 4 (cid:0) X s,t − X s +1 ,t X s − ,t (cid:1) ≥ . The proof of the lemma is completed. (cid:3)
Choosing t = k − in (4.12), it is easy to see that (4.11) holds. Hence we complete the proofof Proposition 4.4. (cid:3) We are now in a position to prove the general case of Theorem 1.5.
N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 21
Proof of Theorem 1.5:
By (4.1) and (4.4), the assumption is equivalent to k X j = l e b j e L j = l − X i =0 e a i e L i , where e a i = (cid:18) n − i (cid:19) (2 i !) a i , e b j = (cid:18) n − j (cid:19) (2 j !) b j . Inductively using (4.9), we get e L j e N i ≥ e N j e L i , for j > i, (4.14)thus we have k − X i =0 e a i e N i k X j = l e b j e L j ≥ k X j = l e b j e N j k − X i =0 e a i e L i . (4.15)Hence k − X i =0 e a i e N i ≥ k X j = l e b j e N j . (4.16)Therefore applying (4.7), we have Z Σ V ( k X j = l e b j e L j − k − X i =0 e a i e L i ) dµ = Z Σ h X, ν i ( k X j = l e b j e N j − k − X i =0 e a i e N i ) dµ ≤ . (4.17)Arguing as in the proof of Theorem 4.3, one can exclude the case (ii) in Proposition 4.4. Hencewe conclude Σ is a geodesic sphere. (cid:3) A suitable adaption of the above argument allows us to demonstrate the same result in thehemisphere case, Theorem 1.6.
Proof of Theorem 1.6:
According to the proof of Theorem 1.5, it suffices to establish the cor-responding inequality of (4.9) in S n + under the assumption of k -convexity. The proof basicallyfollows from the one of Proposition 4.4 except some modifications, so we briefly sketch it here.First, using the simple fact (cid:0) ki (cid:1) = (cid:0) k − i (cid:1) + C i − k − , in view of (4.1), we can split e L k and e N k intotwo terms e L k = k − X i =0 (cid:18) k − i (cid:19) H k − i + k − X i =0 (cid:18) k − i (cid:19) H k − − i , e N k = k − X i =0 (cid:18) k − i (cid:19) H k +1 − i + k − X i =0 (cid:18) k − i (cid:19) H k − − i . Next we introduce the notation X s,t =: t X i =0 (cid:18) ti (cid:19) H s +2 t − i . It is clear that e L k = X ,k − + X ,k − , e L k − = X ,k − , e N k = X ,k − + X ,k − , e N k − = X ,k − . By a similar induction argument as in the proof of Lemma 4.5, one can show thatFor any s ≥ and t ≥ , X s,t X s +1 ,t ≥ X s − ,t X s +2 ,t . (4.18)Finally choosing s = 1 , t = k − in (4.18), we obtain(4.19) e N k − e L k ≥ e N k e L k − . We complete the proof. (cid:3)
In a similar way, one can also prove the rigidity result for the curvature functions N k . Weonly state the result here and leave the proof to readers. Theorem 4.6.
Let ≤ l < k ≤ n − be two integers and Σ be a closed (2 k + 1) -convex ( horospherical convex resp. ) hypersurface embedded in the hyperbolic space S n + ( H n , resp. ) . Ifthere are nonnegative constants { a i } l − i =0 and { b j } kj = l , at least one of them not vanishing, suchthat l − X i =0 a i N i = k X j = l b j N j , then Σ is a centered geodesic hypersphere. In particular, if N k is constant, then Σ is a centeredgeodesic hypersphere. We end this paper with a remark.
Remark 2.
By virtue of our main results, Theorems 1.2, 3.1, 1.5, 1.6 and 4.6, we tend to believethat rigidity holds for hypersurfaces with constant linear combinations of H k , i.e., n − X i =1 a k H k = const. for any a k ∈ R , not necessary nonnegative. In fact, Theorem 1.5, 1.6 and 4.6 include a large classof such rigidity results for linear combinations of H k with pure even (or odd) indices. However,our method seems not enough to prove the most general version of linear combinations. Acknowledgment.
Both authors would like to thank Professors Guofang Wang, Yuxin Ge andDr. Wei Wang for helpful discussions.
N RIGIDITY OF HYPERSURFACES WITH CONSTANT CURVATURE FUNCTIONS 23
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E-mail address : [email protected] Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, D-04103, Leipzig,Germany
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