Spacelike Hypersurfaces in De Sitter Space with Constant Higher-order Mean Curvature

The authors apply the generalized Minkowski formula to set up a spherical theorem. It is shown that a compact connected hypersurface with positive constant higher-order mean curvature H r for some fixed r, 1 ≤ r ≤ n, immersed in the de Sitter space S n+1 1 must be a sphere.


Introduction
The classical Liebmann theorem states that a connected compact surface with constant Gauss curvature or constant mean curvature in R 3 is a sphere.The natural generalizations of the Gauss curvature and mean curvature are the rth mean curvature H r , r = 1,...,n, which are defined as the rth elementary symmetric polynomial in the principal curvatures of M. Later many authors [1,4,5,7,8] have generalized Liebmann theorem to the cases of hypersurfaces with constant higher-order mean curvature in the Euclidian space, hyperbolic space, the sphere, and so on.A significant result due to Ros [8] is that a compact hypersurface with the rth constant mean curvature H r , for some r = 1,...,n, embedded into the Euclidian space must be a sphere.
The purpose of this note is to prove a spherical theorem of the Liebmann type for the compact spacelike hypersurface immersed in the de Sitter space by setting up a generalized Minkowski formula.The main result is the following.
Theorem 1.1.Let M be a compact connected hypersurface immersed in the de Sitter space S n+1 1 .If for some fixed r, 1 ≤ r ≤ n, the rth mean curvature H r is a positive constant on M, then M is isometric to a sphere.
For the cases of the constant mean curvature and constant scalar curvature, that is, r = 1,2, the theorem was founded by Montiel [4] and Cheng and Ishikawa [1], respectively.

Preliminaries
Let R n+2 1 be the real vector space R n+2 endowed with the Lorentzian metric •, • given by for x, y ∈ R n+2 .The de Sitter space S n+1 1 (c) can be defined as the following hyperquadratic: In this way, the de Sitter space inherits from •, • a metric which makes it an indefinite Riemannian manifold of constant sectional curvature c.If x ∈ S n+1 1 (c), we can put 1 be a connected spacelike hypersurface immersed in the de Sitter space with the sectional curvature 1.Following O'Neill [6], the unit normal vector field N for ψ can be viewed as the Gauss map of M: ( Let S r : R n → R, r = 1,...,n, be the normalized rth elementary symmetric function in the variables y 1 ,..., y n .For r = 1,...,n, we denote by C r the connected component of the set {y ∈ R n | S r (y) > 0} containing the vector y = (1,...,1).Notice that every vector (y 1 ,..., y n ) with all its components greater than zero lies in each C r .We derive the following two lemmas, which will be needed for the proof of the theorem.
be a connected spacelike hypersurface immersed in de Sitter space S n+1 1 .For the rth mean curvature H r of ψ, r = 0,1,..., where H 0 = 1 and a ∈ R n+1 1 is an arbitrary fixed vector and N is the unit normal vector of M. Proof.The argument is based on the approach of geodesic parallel hypersurfaces in [5].Let k r and e i , i = 1,...,n, be the principal curvatures and the principal directions at a point p ∈ M. The rth mean curvature of ψ is defined by the identity (2.8) Then the unit normal vector field of ψ t with |N t | 2 = −1 can be written as (2.11) Then the mean curvature H(t) of ψ can be expressed as (2.13) Then we get

.14)
By the way, we must point out that the formula (7 ) in [5] is incorrect because the second term in the right-hand side of the expression of H(t) should be P n (tanht)/nP n (tanht).The volume element dV t for immersion ψ t can be given by where dV is the volume element of ψ.It is an easy computation that ψ,a + H N,a = 0, (2.16) where N is a unit normal field of ψ and a ∈ R n+2 1 an arbitrary fixed vector (cf.[4, page 914]).Integrating both sides of (2.16) over the hypersurface M and applying Stoke's theorem, we get M ψ,a + H 1 N,a dV = 0. (2.17) By using the expression we obtain (2.21) The left-hand side in the equality is a polynomial in the variable tanht.Therefore, all its coefficients are null.This completes the proof of Lemma 2.2.

Proof of Theorem 1.1
Here we work for the immersed hypersurfaces in S n+1 1 instead of embedded hypersurfaces because we can only use algebraic inequalities and the integral formula above to complete the proof.Let some H r be a positive constant.Multiplying (2.17We know from Newton inequality [2] that H r−1 H r+1 ≤ H 2 r , where the equality implies that k Thus we conclude that and if r ≥ 2, the equalities happen only at umbilical points of M. We choose a constant vector a such that |a| If there is a convex point on M, that is, a point at which k i > 0, for all i = 1,...,n, then the constant rth mean curvature H r is positive.By means of the same argument as that of Theorem 1.1, we derive the following.
Corollary 3.1.Let M be a compact connected hypersurface immersed in the de Sitter space S n+1 1 .If for some fixed r, 1 ≤ r ≤ n, the rth mean curvature H r is constant, and there is a convex point on M, then M is isometric to a sphere.
Cai and H. Xu 3 for all t ∈ R. Thus H 1 = H is the mean curvature, H 2 = (n 2 H 2 − S)/n(n − 1), where S is the square length of the second fundamental form and H n is the Gauss-Kronecker curvature of M immersed in S n+11 .Let us consider a family of geodesic parallel hypersurfaces ψ t given by n (2.7) K.