COMPACT SPACE-LIKE HYPERSURFACES IN DE SITTER SPACE

We present some integral formulas for compact space-like hypersurfaces in de Sitter space and some equivalent characterizations for totally umbilical compact space-like hypersurfaces in de Sitter space in terms of mean curvature and higher-order mean curvatures.


Introduction
It is well known that the semi-Riemannian (pseudo-Riemannian) manifolds (M,g) of Lorentzian signature play a special role in geometry and physics, and that they are models of space time of general relativity.Let M n+1 p (c) be an (n + 1)-dimensional complete connected semi-Riemannian manifold with constant sectional curvature c and index p (see [13, page 227]).It is called an indefinite space form of index p and simply a space form when p = 0.According to c > 0, c = 0, and c < 0, M n+1 1 (c) is called de Sitter space, Minkowski space, and anti-de Sitter space, and is denoted by S n+1 1 (c), R n+1 1 , and H n+1 1 (c), respectively.In spite of the fact that the geometry of de Sitter space is the simplest model of space time of general relativity, this geometry was not studied thoroughly.Let φ : M n → S n+1 1 (c) be a smooth immersion of an n-dimensional connected manifold into S n+1 1 (c).If the semi-Riemannian metric of S n+1 1 (c) induces a Riemannian metric on M n via φ, M n is called a space-like hypersurface in de Sitter space.
The study of space-like hypersurfaces in de Sitter space S n+1 1 (c) has been of increasing interest in the last years, because of their nice Bernstein-type properties.Since Goddard [7] conjectured in 1977 that complete space-like hyperspaces in S n+1 1 (c) with constant mean curvature H must be totally umbilical, which turned out to be false in this original statement, an important number of authors have considered the problem of characterizing the totally umbilical space-like hypersurfaces in de Sitter space in terms of some appropriate geometric assumptions.Actually, Akutagawa [1] proved that Goddard's conjecture is true when H 2 ≤ c if n = 2, and H 2 < (4(n − 1)/n 2 )c if n ≥ 3. On the other hand, Montiel [11] proved that Goddard's conjecture is also true under the additional hypothesis of the compactness of the hypersurfaces.We also refer to [14] for an alternative proof of both facts given by Ramanathan in the 2-dimensional case.More recently, Cheng and Ishikawa [5] have shown that compact space-like hyperspaces in S n+1 1 (c) with constant 2054 Compact space-like hypersurfaces in de Sitter space scalar curvature S < n(n − 1)c must be totally umbilical.Aledo el al. [3] have recently found some other characterizations of the totally umbilical compact space-like hypersurfaces in de Sitter space with constant higher-order mean curvatures, under appropriate hypothesis.
In this paper, we will study various equivalent characterizations of totally umbilical compact space-like hypersurfaces in de Sitter space in terms of mean curvature and higher-order mean curvatures.The whole paper is organized as follows.Section 2 gives some preliminaries, Section 3 gives some inequalities on the normalized symmetric functions, and Section 4 reviews some selfadjoint second-order differential operator.The main results of this paper are contained in Section 5, which gives us a more specific and complete picture of totally umbilical compact space-like hypersurfaces in de Sitter space.For simplicity, we omit the volume form dV in all integrals.

Preliminaries
We consider Minkowski space R n+2 1 as the real vector space R n+2 endowed with the Lorentzian metric •, • given by for x, y ∈ R n+2 .Then de Sitter space S n+1 1 (c) can be defined as the following hyperquadric of R n+2 1 : 2) The induced metric from •, • makes S n+1 1 (c) into a Lorentzian manifold with constant sectional curvature c.Moreover, if x ∈ S n+1 1 (c), we can put We denote by ∇ L and ∇ the metric connections of R n+2 1 and S n+1 1 (c), respectively.Then, we have be a space-like hypersurface in S n+1 1 (c) defined above.First, we want to know whether a compact one is orientable.The following proposition gives us the affirmative answer (see [11] or [2] for a proof).

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Throughout the following, we will exclusively deal with compact space-like hypersurfaces in S n+1 1 (c), n ≥ 2. The above proposition ensures that M n is orientable.Let N be a time-like unit normal vector field for the immersion φ.The field N can be viewed as the Gauss map of M n into hyperbolic space: where x 0 ≥ 1}.We will say that M n is oriented by N. A well-known result is that the Gauss map N is harmonic if and only if the mean curvature H is constant.For a proof, one can refer to [4].
Let ∇ be the Levi-Civita connection associated to the Riemannian metric on M n induced from •, • .Then, we have where Ꮽ stands for the shape operator of the immersion φ with respect to N and v,w are vector fields tangent to M n .The operator L = −Ꮽ is the Weingarten endomorphism.The eigenvalues of the operator L are called the principal curvatures and will be denoted by λ 1 ,...,λ n .The Codazzi equation is expressed by For a suitably chosen local field of orthonormal frames e 1 ,...,e n on M n , we have (2.9) The kth mean curvature of the space-like hypersurface M n is defined by (2.10) Note that when k = 1, H 1 is the mean curvature H, and when k = n, H n is the Gauss-Kronecker curvature.We can easily see that the scalar curvature and the characteristic polynomial of Ꮽ can be written in terms of the H k 's as where H 0 = 1.
2056 Compact space-like hypersurfaces in de Sitter space Minkowski formulas provide us with a convenient tool in the study of hypersurfaces.One can refer to [12] for the well-known version for space forms.Many interesting results have been got in the study of hypersurfaces by means of Minkowski formulas, for example, [9,10,12,16,17], and so forth.The proof in [12] followed the idea in [15].Similar to it, one can easily give the proof of Minkowski formulas for compact space-like hypersurfaces in de Sitter space (see [3]).The following proposition is Minkowski formulas for compact space-like hypersurfaces in de Sitter space.

Inequalities on the normalized symmetric functions
Let x 1 ,...,x n ∈ R. The elementary symmetric functions of n variables x 1 ,...,x n are defined by where σ 0 = 1.For our purpose, it is useful to consider the normalized symmetric functions by dividing each σ k by the number of its summands.We denote the normalized symmetric function by where E 0 = 1.Since we see that at least r of x i 's are zero if and only if Proposition 3.1.All x i ≥ 0 if and only if all E i ≥ 0, and all x i > 0 if and only if all E i > 0.
Proof.We prove it by induction on n.For n = 1, the proposition holds clearly.Now assume that n > 1 and the proposition holds for n . By Rolle's theorem, y 1 ,..., y n−1 are all real and Clearly, the inductive assumption applies to y 1 , ..., y n−1 .Thus, it follows easily that the proposition holds for n.
There are some well-known inequalities on the normalized symmetric functions, for example, Newton-Maclaurin inequalities.One can refer to [8] for the case of n positive numbers.For the sake of completeness, we include here a proof of Newton's inequalities for the general case.
and each equality holds if and only if Proof.We prove it by induction on n.For n = 2, the inequality holds clearly and the equality holds if and only if Now assume that n > 2 and the proposition holds for n − 1.Let On the other hand, where y 1 ,..., y n−1 are n − 1 roots of the polynomial Q(x).Comparing the coefficients of the powers of x in the above two expressions for Q(x) gives us By Rolle's theorem y 1 ,..., y n−1 are all real.Clearly, Thus the inductive assumption applies to E i (y 1 ,..., y n−1 ), i = 0,...,n − 1, and the proposition holds for k = 1,...,n − 2 by (3.7).It remains to prove for k = n − 1, that is, with equality if and only if Clearly, (3.8) holds with equality if and only if E n−1 = (1/n) j =i x j = 0, and thus if and only if some x j = 0, j = i.
Case 2. If all x i = 0, let x i = 1/x i .Then, we have which is true since n > 2.
2058 Compact space-like hypersurfaces in de Sitter space This completes the proof.
Remark 3.3.For our future purpose, we concern most when each of the above equalities holds if and only if x n , that is, to find some restrictions on x i 's to exclude the possibility of E k = 0 = E k−1 E k+1 and x i 's are not all zero.We only know that E 2 1 = E 2 holds if and only if while we cannot expect it for k ≥ 2 even if all x i ≥ 0, for example, when only one of x i 's is positive.In particular, when all x i 's have the same sign, that is, nonnegative or nonpositive simultaneously, and at least k of x i 's are nonzero (equivalently, Newton's inequalities have a very important consequence, Maclaurin's inequalities, by investigating that where all x i ≥ 0. When all x i > 0 and 2 ≤ k ≤ n − 1, we have with equality if and only if If some of x i 's are zero and the rest of them are positive, then for 2 ≤ k ≤ n − 1, we still have with equality if and only if with equality if and only if with each equality if and only if Now we can give a result on the positiveness of mean curvature and higher-order mean curvatures of the compact space-like hypersurfaces in de Sitter space. Theorem 3.6.Let φ : M n → S n+1 1 (c), n ≥ 2, be a compact space-like hypersurface in de Sitter space with H k > 0 and 2 ≤ k ≤ n.If there exists a point of M n , where H 1 ,...,H k−1 are positive, then H 1 ,...,H k−1 are positive everywhere on M n , that is, H 1 > 0,...,H k−1 > 0.

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Proof.We prove it by an open-closed argument.Let Clearly U is open, and it is nonempty by the assumption.To prove that U = M n , we only need to prove that U is also closed by the connectedness of M n .Since H k > 0 and M n is compact, we have (3.17) For any x ∈ U, we have by Corollary 3.5.Thus U is closed.This completes the proof.
Finally, we give another two sets of important inequalities by investigating that where all x i ≥ 0 and 1 ≤ k < l ≤ n − 1.Using the argument above leading to Corollary 3.4, we can get the following important inequalities.
Theorem 3.7.If all x i ≥ 0 and with equality if and only if with equality if and only if Theorem 3.9.If all x i ≥ 0 and with equality if and only if with equality if and only if 2060 Compact space-like hypersurfaces in de Sitter space

Some selfadjoint second-order differential operators
First, we introduce two known selfadjoint second-order differential operators, the Laplace operator and the Cheng-Yau operator .For any C 2 -function f defined on M n , we consider the symmetric bilinear form The Laplace operator acting on any C 2 -function f defined on M n is given by Since M n is compact and oriented, the Laplace operator is selfadjoint relative to the L 2 -inner product of M n , that is, Following Cheng and Yau [6], we introduce an operator acting on any C 2 -function f defined on M n by Note that the following holds at umbilical points: By the Codazzi equation and [6, Proposition 1], we can prove that the operator is selfadjoint relative to the L 2 -inner product of M n , that is, Naturally, we may ask the following question.
Question 4.1.Can we find other selfadjoint second-order differential operators in terms of the shape operator Ꮽ, mean curvature, and higher-order mean curvatures?
Fortunately, we do have such a selfadjoint second-order differential operator ᏸ k for each k = 0,1,...,n − 1.The idea is contained in [15,17].Following [3], we introduce the kth Newton transformation T k associated to the shape operator Ꮽ: Jinchi Lv 2061 or inductively, It follows from (2.12) that T n = 0. Since the shape operator Ꮽ is selfadjoint, it follows easily that the Newton transformations T k 's are selfadjoint.Clearly, the orthonormal basis {e 1 ,...,e n } diagonalizes the Newton transformations T k 's since it diagonalizes the shape operator Ꮽ.
Proof.Since the shape operator Ꮽ is negative definite, all λ i > 0. Without loss of generality, to prove that T k is positive definite, we only need to prove that T k e 1 ,e 1 > 0. Let λ i = λ i /λ 1 , i = 1,...,n, then we have Let n = m and k = l, then we have by the inductive assumption and the fact that k i=0 (−1) k−i σ i (1,x 2 ,...,x n−1 ) = 0 for k = n − 1.This completes the proof.
2062 Compact space-like hypersurfaces in de Sitter space The following algebraic properties of T k can be easily established from the definitions.
One can also easily derive the identities where v is any vector field tangent to M n .Now for each k = 0,1,...,n − 1, we can define a second-order differential operator ᏸ k acting on any C 2 -function f defined on M n by It can be easily seen that the operators ᏸ k 's are selfadjoint.Clearly when k = 0, the operator ᏸ 0 is the Laplace operator = div •∇.Later, we will see that when k = 1, the operator ᏸ 1 is the Cheng-Yau operator .Finally, we can easily derive the following useful expression for ᏸ k (see [3]): T k e i ,e i ∇ 2 f e i ,e i (4.17) for any C 2 -function f defined on M n .

.18)
Clearly when k = 1, the operator ᏸ 1 is the Cheng-Yau operator = i (nH − λ i )∇ 2 .Note that the following holds at umbilical points: Remark 4.4.When T k is positive definite, the operator ᏸ k is elliptic.In particular, when the shape operator Ꮽ is negative definite, the operator ᏸ k is elliptic by proposition 4.2.

Main results
Let φ : M n → S n+1 1 (c), n ≥ 2, be a compact space-like hypersurface in de Sitter space, N a time-like unit normal vector field for φ, and a ∈ R n+2 1 arbitrary.We consider the height function φ,a and the function N,a on M n .Using (2.4), (2.7), we can get the following expressions for the gradient and Hessian of the above two functions: where v,w are vector fields tangent to M n .Thus, we have ( Note that the Minkowski formulas in Proposition 2.2 are regained by the selfadjointness of the operators ᏸ k 's.
2064 Compact space-like hypersurfaces in de Sitter space For any vector field v tangent to M n , we have by the selfadjointness of the operator ∇ v Ꮽ, where a T is the tangent component of a to M n .Thus by (2.8), (4.13), (4.14), and (4.15), we have (5.4) Remark 5.1.In particular, when k = 0, we have ) Proof.By (5.2), we have (5.7)
2066 Compact space-like hypersurfaces in de Sitter space Theorem 5.4.Let φ : M n → S n+1 1 (c), n ≥ 2, be a compact space-like hypersurface in de Sitter space, a ∈ R n+2 1 any unit time-like vector with the same time-orientation as N, and 0 and the equality holds if and only if M n is totally umbilical when k = 0, or additionally if For any unit time-like vector a ∈ R n+2 1 with the same time orientation as N, that is, |x| 2 = −1 and x 0 ≥ 1, we have N,a ≤ −1.Thus by taking i = k, j = k + 1 in Theorem 5.3 and Proposition 3.2, we can deduce that and the equality holds if and only if M n is totally umbilical when k = 0 or additionally if and the equality holds if and only if M n is totally umbilical for any unit time-like vector a ∈ R n+2 1 with the same time orientation as N.
Remark 5.6.In particular, if H k and H k+1 are constant, 0 ≤ k ≤ n − 2, then M n is totally umbilical when k = 0, or additionally if See also [3].
Theorem 5.7.Let φ : M n → S n+1 1 (c), n ≥ 2, be a compact space-like hypersurface in de Sitter space with H 1 ≥ 0,...,H n ≥ 0, a ∈ R n+2 1 any unit time-like vector with the same time orientation as N, and with the same time orientation as N.By Theorems 5.4 and 5.7, we have we have Thus, M n is totally umbilical by Theorem 5.7.
Theorem 5.9.Let φ : M n → S n+1 1 (c), n ≥ 2, be a compact space-like hypersurface in de Sitter space with H k+1 > 0, a ∈ R n+2 1 any unit time-like vector with the same time orientation as N, and x,a = 0} defines an n-sphere which is a totally geodesic hypersurface in S n+1 1 (c).We will refer to that sphere as the equator of S n+1 1 (c) determined by a.This equator divides the de Sitter space into two connected components; the future which is given by x,a < 0 , (5.25) 2068 Compact space-like hypersurfaces in de Sitter space and the past given by x,a > 0 . (5.26) Following [3], we can easily get the following corollary.
Corollary 5.10.Let φ : M n → S n+1 1 (c), n ≥ 2, be a compact space-like hypersurface in de Sitter space and 2 ≤ k ≤ n − 1.If M n is contained in the chronological future (or past) relative to the equator of S n+1 1 (c) determined by a unit time-like vector a ∈ R n+2 1 with the same time orientation as N and H k+1 > 0 (or (−1) k+1 H k+1 > 0), then ∇H i ,a ≥ 0 , 2≤ i ≤ k, (5.27) with each equality if and only if M n is totally umbilical.
Proof.First we prove the future case.By Theorem 5.9, it is sufficient to prove that there exists a point of M n , where all H i > 0. Since M n is contained in the chronological future relative to the equator determined by a and M n is compact, there exists a point x 0 ∈ M n such that max x∈M n φ(x),a = φ x 0 ,a < 0. (5.28) Thus by maximum principle, we have −c φ x 0 ,a + λ i N x 0 ,a = −c e i ,e i φ x 0 ,a − Ꮽe i ,e i N x 0 ,a = ∇ 2 φ,a e i ,e i ≤ 0. (5.29) Since a ∈ R n+2 1 is a unit time-like vector with the same time orientation as N, we have N,a ≤ −1.So λ i ≥ c φ(x 0 ),a N x 0 ,a > 0, i = 1,...,n.
(5.30) Thus all H i > 0. For the past case, we only need to replace N and a by −N and −a, respectively, and the proof for the future case applies.This completes the proof.
Corollary 5.8.Let φ : M n → S n+1 1 (c), n ≥ 2, be a compact space-like hypersurface in de Sitter space with H 1 ≥ 0,...,H n ≥ 0 and constant k−1 i=1 a i k = 0, the equality holds if and only if M n is totally umbilical.
there exists a point of M n , where H 1 ,...,H k are positive, then with equality if and only if M n is totally umbilical.Proof.For any unit time-like vector a ∈ R n+2 1 with the same time orientation as N, we have N,a ≤ −1.Thus by Theorems 5.3, 3.6, and 3.8, we can deduce that