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 $\mathbb H_1^{n+1}$. 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 $\mathbb H_1^{n+1}$. 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).