Rigidity and non-existence of spacelike submanifolds with causal mean curvature vector field in spacetimes and the Cauchy problem in General Relativity

2019/11/10 by uan A. Aledo, Aledo, uan A., Rafael M. Rubio +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #math-ph #math.DG #math.MP

paper · pdf · doi:10.48550/arxiv.1911.03928

arxiv created 2019/11/10 · openalex publication_date 2019/11/10 · arxiv updated 2019/11/12 · openalex created_date 2019/11/22 · openalex updated_date 2026/07/28

Abstract

New general results of non-existence and rigidity of spacelike submanifolds immersed in a spacetime, whose mean curvature is a time-oriented causal vector field, are given. These results hold for a wide class of spacetimes which includes globally hyperbolic, stationary, conformally stationary and pp-wave spacetimes, among others. Moreover, applications to the Cauchy problem in General Relativity, are presented. Finally, in the case of hypersurfaces, we also obtain significant consequences in Geometrical Analysis, solving new Calabi-Bernstein and Dirichlet problems on a Riemannian manifold.

Citations

• Trapped surfaces, horizons and exact solutions in higher dimensions
• The singularities of gravitational collapse and cosmology
• Curvature estimates for minimal hypersurfaces

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