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

Abstract

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