Compact space-like hypersurfaces in de Sitter space

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 Mn+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 1 (c) is called de Sitter space, Minkowski space, and anti-de Sitter space, and is denoted by S 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 :Mn → S 1 (c) be a smooth immersion of an n-dimensional connected manifold into S 1 (c). If the semiRiemannian metric of S 1 (c) induces a Riemannian metric on M n via φ, Mn is called a space-like hypersurface in de Sitter space. The study of space-like hypersurfaces in de Sitter space S 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 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 H2 ≤ c if n= 2, and H2 < (4(n− 1)/n2)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 1 (c) with constant