SPACELIKE HYPERSURFACES WITH

CONSTANT SCALAR CURVATURE*

Qing-Ming Cheng & Susumu Ishikawa

Abstract. In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y.Cheng and S.T.Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S_{1}^{n+1}(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n − 1)r is isometric to a sphere if r < c.

1. Introduction

Let M_{1}^{n+1}(c) be an (n+1)-dimensional connected indefinite Riemannian manifold of index 1 and of constant curvature c. According to c > 0, c = 0 and c < 0, it is called de Sitter space, Minkowski space or anti-de Sitter space, respectively, and it is denoted by S_{1}^{n+1}(c), R_{1}^{n+1} or H_{1}^{n+1}(c). A hypersurface M of M_{1}^{n+1}(c) is said to be spacelike if the induced metric on M from that of the ambient space is positive definite. For indefinite Riemannian manifolds, refer to B.O′Neill [11]. E. Calabi [2] proved that a maximal spacelike entire graph in the Minkowski space R_{1}^{n+1} with n ≤ 4 is a linear hyperplane. Later, S.Y.Cheng and S.T.Yau [6] showed that the same holds for arbitrary n. Since the importance of spacelike hypersurfaces in general relativity has been emphasized by a number of authors, complete spacelike hypersurfaces have been studied by many mathematicians.

By studying the Laplacian of certain globally geometric invariants (for examples, the Laplacian of the squared norm of the second fundamental form, and so on), Bernstein type theorems of complete spacelike hypersurfaces with constant mean curvature have been obtained (e.g. [1], [3], [4], [5], [8] and so on). For the investigation of hypersurfaces with constant scalar curvature, the method does not work very well. In our research, we shall make use of a self-adjoint operator □ introduced by S.Y.Cheng and S.T.Yau to study compact hypersurfaces with constant scalar curvature n(n − 1)r immersed in de Sitter space. Recently, Y.Zheng [12] and [13] (cf. Cheng [3]) proved the following theorem:

Theorem Z. Let ϕ : M → S_{1}^{n+1}(c) be an n-dimensional compact spacelike hypersurface with constant scalar curvature immersed in the de Sitter space S_{1}^{n+1}(c). If

K(M) ≥ 0, r < c,

Typeset by AMS-TEX

Key words and phrases. spacelike hypersurface, scalar curature and de Sitter space.

*Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.

1991 Mathematics Subject Classification 53C20, 53C42

then M is isometric to a sphere, where K(M) and n(n − 1)r are the sectional curvature and the scalar curvature of M, respectively.

In this paper, we shall give an integral equality for compact spacelike hypersurfaces with constant scalar curvature immersed in the de Sitter space S_{1}^{n+1}(c). By making use of this integral equality, we shall prove the following:

Theorem. Let ϕ : M → S_{1}^{n+1}(c) be an n-dimensional compact spacelike hypersurface with constant scalar curvature immersed in the de Sitter space S_{1}^{n+1}(c). If r < c, then M is isometric to a sphere, where n(n − 1)r is the scalar curvature of M.

Acknowledgement. The first named author would like to thank professor K. Shiohama for his constant encouragement and help. The authors also would like to express their gratitude to the referee for his valuable suggestions.

2. Preliminaries

We consider Minkowski space R_{1}^{n+2} as the real vector space R^{n+2} endowed with the Lorentzian metric ⟨, ⟩ given by

(2.1)  ⟨x, y⟩ = −x_{0}y_{0} + ∑_{i=1}^{n+1} x_{i}y_{i}

for x, yϵR^{n+2}. Then, for c > 0, the de Sitter space S_{1}^{n+1}(c) can be defined as the following hyperquadric of R_{1}^{n+2}

S_{1}^{n+1}(c) = \{xϵR_{1}^{n+2}; |x|^{2} = \frac{1}{c}\}.

In this way, the de Sitter space S_{1}^{n+1}(c) inherits from ⟨, ⟩ a metric which makes it an indefinite Riemannian manifold of constant sectional curvature c. For indefinite Riemannian manifolds, refer to B. O′Neill [11].

Moreover, if x ∈ S_{1}^{n+1}(c), we can put T_{x}S_{1}^{n+1}(c) = {v ∈ R_{1}^{n+2}|⟨v, x⟩ = 0}. If ∇^{L} and ¯∇ denote the metric connections of R_{1}^{n+2} and S_{1}^{n+1}(c) respectively, we have

(2.2)  ∇_{v}^{L}w − ¯∇_{v}w = −c⟨v, w⟩x

for all vector fields v, w which are tangent to S_{1}^{n+1}(c). Let

(2.3)  ϕ : M → S_{1}^{n+1}(c)

be a connected spacelike hypersurface immersed in S_{1}^{n+1}(c). Following B.O′Neill [11], the unit normal vector field N for ϕ can be viewed as the Gauss map of M:

(2.4)  N : M → {x ∈ R_{1}^{n+2}; |x|^{2} = −1}.

Since {x ∈ R_{1}^{n+2}; |x|^{2} = −1} has two connected components, say x_{0} ≥ 1 and x_{0} ≤ −1, we shall choose N to satisfy

(2.5)  N(M) ⊂ {x ∈ R_{1}^{n+1}; |x|^{2} = −1, x_{0} ≥ 1}.

Let ∇ be the Levi-Civita connection corresponding to the Riemannian metric g induced on M from ⟨, ⟩. Then the second fundamental form h⃗ and the Weingarten endomorphism A of ϕ are given by

(2.6)  ¯∇_{v}w − ∇_{v}w = h⃗(v, w),

(2.7)  ¯∇_{v}N = −Av and h⃗(v, w) = −g(Av, w)N,

where v, w are vector fields tangent to M. So, the mean curvature H of the immersion ϕ is given by nH = traceA. Let R denote the curvature tensor field of M. The Gauss equation is:

(2.8)  R(v, w)y = c[g(w, y)v − g(v, y)w] − [g(Aw, y)Av − g(Av, y)Aw],

where v, w and y are vector fields tangent to M. The Codazzi equation is expressed by

(2.9)  (∇_{v}A)w = (∇_{w}A)v.

We take a local field of orthonormal frames e_{1}, ⋯, e_{n} on M. From (2.8), we have

(2.10)  n(n − 1)(r − c) = S − (nH)^{2},

where S = |h⃗|^{2} and n(n−1)r are the squared norm of the second fundamental form and the scalar curvature of M, respectively.

For a C^{2}-function f defined on M, we consider the symmetric bilinear form

(2.11)  (∇^{2}f)(w, v) = v(wf) − (∇_{v}w)f.

The Laplace operator Δ acting on any C^{2}-function defined on M is given by

(2.12)  Δf = ∑_{j=1}^{n}(∇^{2}f)(e_{j}, e_{j}).

Following S.Y.Cheng and S.T. Yau [7], we introduce an operator □ acting on any C^{2}-function f defined on M by

(2.13)  □f = ∑_{i,j=1}^{n}[nHg(e_{i}, e_{j}) − g(Ae_{i}, e_{j})](∇^{2}f)(e_{i}, e_{j}).

Making use of the Codazzi equation (2.9), we can prove that the operator □ is self-adjoint relative to the L^{2} inner product of M, that is,

∫_{M} f(□g) = ∫_{M}(□f)g, (cf. the Proposition 1 in [7]).

We choose a local field of orthonormal frames e_{1}, ⋯, e_{n} on M such that, at p,

(2.14)  Ae_{i} = λ_{i}e_{i},

where {λ_{i}} are the principal curvatures of M.

Lemma. Let ϕ : M → S_{1}^{n+1}(c) be an n-dimensional spacelike hypersurface with constant scalar curvature n(n−1)r immersed in a de Sitter space S_{1}^{n+1}(c). If r < c, then the operator □ is an elliptic operator.

Proof. Since the scalar curvature is constant and r < c, we have (nH)^{2} > S from (2.12). Therefore, we can assume that the mean curvature is positive. If there exists a λ_{i} such that

nH − λ_{i} ≤ 0,

then (nH)^{2} ≤ λ_{i}^{2} ≤ S. This is a contradiction. Hence, nH − λ_{i} > 0 for any i. From the definition of the operator □, we know that □ is an elliptic operator if r < c.

3. Proof of theorem

Firstly, we prove the following integral equality:

Proposition. Let ϕ : M → S_{1}^{n+1}(c) be an n-dimensional compact spacelike hypersurface with constant scalar curvature immersed in the de Sitter space S_{1}^{n+1}(c). Then, for any vector a in R^{n+2}, we have the integral equality

∫_{M}[(n − 2)HB − B_{3}]⟨N, a⟩ = 0.

Here B = S − nH^{2} and B_{3} = ∑_{i}(λ_{i} − H)^{3}.

Proof. We consider the functions ⟨N, a⟩ and ⟨ϕ, a⟩ on M. From (2.2), (2.6) and (2.7), we have

v⟨ϕ, a⟩ = ⟨v, a⟩, v⟨N, a⟩ = −⟨Av, a⟩,

and

(3.1)  (∇^{2}⟨ϕ, a⟩)(v, w) = wv⟨ϕ, a⟩ − (∇_{w}v)⟨ϕ, a⟩
= −c⟨v, w⟩⟨ϕ, a⟩ − ⟨Av, w⟩⟨N, a⟩,

(∇^{2}⟨N, a⟩)(v, w) = wv⟨N, a⟩ − (∇_{w}v)⟨N, a⟩ (3.2)

= c⟨Av, w⟩⟨ϕ, a⟩ + ⟨Av, Aw⟩⟨N, a⟩ − ⟨(∇_{w}A)v, a⟩

where v and w are vector fields tangent to M.

We take a local field of orthonormal frames e_{1}, ⋯, e_{n} on M such that (2.14) holds. Then, from (3.1) and (3.2), we have

□⟨N, a⟩ = ∑_{i}(nH − λ_{i})(∇^{2}⟨N, a⟩)(e_{i}, e_{i}) (3.3)

= ((nH)^{2} − S)c⟨ϕ, a⟩ + (nHS − ∑_{i} λ_{i}^{3})⟨N, a⟩

− \frac{1}{2}⟨grad((nH)^{2} − S), a⟩,

and

(3.4)  Δ⟨ϕ, a⟩ = ∑_{i=1}^{n}(∇^{2}⟨ϕ, a⟩)(e_{i}, e_{i}) = −nc⟨ϕ, a⟩ − nH⟨N, a⟩.

Since the scalar curvature n(n − 1)r is constant, we obtain grad(n^{2}H^{2} − S) = 0 from (2.10). Therefore, (3.3) and (3.4) yield

□⟨N, a⟩ + \frac{(nH)^{2} − S}{n} Δ⟨ϕ, a⟩ (3.5)

= {−[(nH)^{2} − S]H + nHS − ∑_{i} λ_{i}^{3}}⟨N, a⟩

= [(n + 1)HS − n^{2}H^{3} − ∑_{i} λ_{i}^{3}]⟨N, a⟩.

Since

(3.6)  ∑_{i} λ_{i}^{3} = B_{3} + 3HB + nH^{3},

(3.5) and (3.6) imply

□⟨N, a⟩ + \frac{(nH)^{2} − S}{n} Δ⟨ϕ, a⟩
= [(n − 2)HB − B_{3}]⟨N, a⟩.

Since the operator □ and the Laplace operator Δ are self-adjoint, we have using Stokes’ formula

∫_{M}[(n − 2)HB − B_{3}]⟨N, a⟩ = 0.

This completes the proof of Proposition. □

In order to prove our Theorem, the following Sublemma due to M. Okumura will be used:

Sublemma (cf. Okumura [9]). Let a_{i} (i = 1, 2, ⋯, n) be the real numbers satisfying ∑_{i} a_{i} = 0 and ∑_{i} a_{i}^{2} = B. Then

−\frac{n − 2}{\sqrt{n(n − 1)}}B^{\frac{3}{2}} ≤ ∑_{i} a_{i}^{3} ≤ \frac{n − 2}{\sqrt{n(n − 1)}}B^{\frac{3}{2}}

and the equality holds if and only if at least (n − 1) of the a_{i} are equal.

Proof of Theorem. We choose the constant vector a in our Proposition such that |a|^{2} = −1 and a_{0} ≤ −1. Since the Gauss map N of M satisfies |N|^{2} = −1 and N_{0} ≥ 1, we have ⟨N, a⟩ ≥ 1 on M. From r < c and (2.10), we have S < (nH)^{2}. Hence H ≠ 0. We can assume H > 0. From the Sublemma mentioned above, we obtain the inequality

(3.7)  (n − 2)HB − B_{3} ≥ (n − 2)HB − \frac{n − 2}{\sqrt{n(n − 1)}}B^{\frac{3}{2}} .

Because of B = S −nH^{2} < n(n−1)H^{2}, the right hand side of (3.7) is non-negative. Hence we obtain

(3.8)  (n − 2)HB − B_{3} ≥ 0.

Therefore, our Proposition and ⟨N, a⟩ ≥ 1 imply B ≡ 0 on M. Thus M is totally umbilical and M is isometric to a sphere. □

References

1. K. Akutagawa, Spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196 (1987), 13-19.

2. E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Symp. Pure Appl. Math. 15 (1970), 223-230.

3. Q.M. Cheng, Complete spacelike hypersurfaces of a de Sitter space with r = kH, Mem. Fac. Sci., Kyushu Univ. 44 (1990), 67-77.

4. Q.M. Cheng, Hypersurfaces of a Lorentz space form, Arch. Math. 63 (1994), 271-281.

5. Q.M. Cheng and H. Nakagawa, totally umbilical hypersurfaces, Hiroshima J. Math. 20 (1990), 1-10.

6. S.Y. Cheng and S.T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. 104 (1976), 407-419.

7. S.Y. Cheng and S.T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195-204.

8. S. Montiel, An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (1988), 909-917.

9. M. Okumura, Hypersurfaces and pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207-213.

10. J. Ramanathan, Complete spacelike hypersurfaces of constant mean curvature in de Sitter space, Indiana Univ. Math. J. 36 (1987), 349-359.

11. B. O′Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, London, 1983.

12. Y. Zheng, Space-like hypersurfaces with constant scalar curvature in the de Sitter space, Diff. Geom. Appl. 6 (1996), 51-54.

13. Y. Zheng, On space-like hypersurfaces in the de Sitter space, Ann. of Global Analysis and Geom. 13 (1995), 317-321.

14. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968), 62-105.

Department of Mathematics, Faculty of Science, Josai University, Saitama, Sakado 350-02, Japan e-mail: cheng@math.josai.ac.jp

Department of Mathematics, Saga University, Saga 840, Japan e-mail: Ishikawa@ms.saga-u.ac.jp