COMPLETE SPACELIKE HYPERSURFACES IN THE ANTI-DE SITTER SPACE: RIGIDITY, NONEXISTENCE AND CURVATURE ESTIMATES

Our purpose is to investigate the geometry of complete spacelike hypersurfaces immersed in the anti-de Sitter space H-1(n+1). We start by proving rigidity results for such hypersurfaces under suitable constraints on their higher order mean curvatures. We also obtain a lower estimate for the index of minimum relative nullity for r-maximal spacelike hypersurfaces and a nonexistence result for 1-maximal spacelike hypersurfaces of H-1(n+1). Finally, we employ a technique due to Aledo and Alias (2000) to prove some curvature estimates for complete spacelike hypersurface of H-1(n+1); as a consequence, we get further nonexistence results. In particular, we show the nonexistence of complete maximal spacelike hypersurfaces in certain open regions of Hn+1 1. Our approach is mainly based on a suitable extension of the generalized maximum principle of Omori and Yau due to Alias, Impera and Rigoli (2012).