Complete spacelike hypersurfaces in the anti-de Sitter space: rigidity, nonexistence and curvature estimates

Weiller F. C. Barboza, Henrique F. de Lima, Arlandson M. S. Oliveira, Marco Antonio L. Velásquez Colloquium Mathematicum MSC: Primary 53C42; Secondary 53B30, 53C50. DOI: 10.4064/cm8502-12-2021 Published online: 23 May 2022

Abstract

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).

Authors

  • Weiller F. C. BarbozaDepartamento de Matemática
    Universidade Federal de Campina Grande
    58.109-970 Campina Grande, Paraíba, Brazil
    e-mail
  • Henrique F. de LimaDepartamento de Matemática
    Universidade Federal de Campina Grande
    58.109-970 Campina Grande, Paraíba, Brazil
    e-mail
  • Arlandson M. S. OliveiraCentro de Ciências
    Exatas e Sociais Aplicadas
    Universidade Estadual da Paraíba
    58.706-550 Patos, Paraíba, Brazil
    e-mail
  • Marco Antonio L. VelásquezDepartamento de Matemática
    Universidade Federal de Campina Grande
    58.109-970 Campina Grande, Paraíba, Brazil
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image