Complete spacelike hypersurfaces in the anti-de Sitter space: rigidity, nonexistence and curvature estimates
Volume 170 / 2022
Abstract
Our purpose is to investigate the geometry of complete spacelike hypersurfaces immersed in the anti-de Sitter space . 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 \mathbb H_1^{n+1}. Finally, we employ a technique due to Aledo and Alías (2000) to prove some curvature estimates for complete spacelike hypersurface of \mathbb 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 \mathbb {H}_{1}^{n+1}. Our approach is mainly based on a suitable extension of the generalized maximum principle of Omori and Yau due to Alías, Impera and Rigoli (2012).
