Hypersurfaces In De Sitter Space Research Articles

Integral formulas for compact space-like hypersurfaces in de Sitter space: Applications to the case of constant higher order mean curvature

In this paper we develop some integral formulas for compact space-like hypersurface in de Sitter space S 1 n+1 and apply them in order to characterized thetotally umbilical round spheres of S 1 n+1 as the only compact space-like hypersurfaces with constant higher order mean curvature under some appropriate hypothesis. In particular, for hypersurfaces contained in the chronological future (or past) of an equator of S 1 n+1 we prove that the only compact space-like hypersurface with a constant higher order mean curvature are the totally umbilical round spheres.

On the Ricci Curvature of Compact Spacelike Hypersurfaces in de Sitter Space

In this paper we establish a sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for the Ricci curvature. Our result will be a consequence of an integral formula involving the Ricci curvature and the scalar curvature of the hypersurface. We also derive some other consequences and applications of this formula.

Integral formulas for compact spacelike hypersurfaces in de sitter space and their applications to Goddard's conjecture

We establish integral formulas of Minkowski's type for compact spacelike hypersurfaces in de sitter spaceS 1 n+1 (1) and give their applications to the case of constantr-th mean curvature (r=1,2,…,n−1). Whenr=1 we recover Montiel's result.

A Priori Estimates of the Principal Curvatures of Spacelike Hypersurfaces in de Sitter Space with Applications to Hypersurfaces in Hyperbolic Space

Introduction. In the last two decades there has been a substantial interest in studying spacelike hypersurfaces with constant mean curvature in Lorentz-Minkowski space R In +1 and in de Sitter space Sin +1, n -2. This interest is motivated by certain needs in general relativity [CFM], [G], [St], and also by the fact that complete maximal and constant mean curvature hypersurfaces in these spaces exhibit the property, that is, they are of a very special type; for example linear subspaces, totally umbilic, or totally geodesic; [Ca], [CY], [G], [R], [A], [M]. It is a natural and interesting problem to investigate stability of a Bernstein-type property relative to perturbation of the data. One of the main purposes of this paper is to give estimates of the spread of the principal curvatures on a compact spacelike hypersurface in SIn+l in terms of the mean curvature and its derivatives (up to second order). Using these estimates we show that if the mean curvature H on a complete spacelike hypersurface F in SIn+ I is such that H2 < 4 (n 1)/n2 and H is close to a constant (in C2-norm) then F is close to being umbilic. A simple consequence of our estimates is the result of Akutagawa [A] and Montiel [M] showing that complete spacelike hypersurfaces in SI' with constant mean curvature H such that H2 < 4 (n 1)/n2 are umbilic. Another aim of this paper is to study duality relations between hypersurfaces in hyperbolic space Hn+l and spacelike hypersurfaces in SIn+l. Apart from the well known fact that the Gauss image of a hypersurface in Hn+l lies in SIn+1 (when the two spaces are considered as