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

TL;DR

This paper establishes new non-existence and rigidity results for spacelike submanifolds with causal mean curvature in various spacetimes, with applications to the Cauchy problem in General Relativity and geometric analysis.

Contribution

It provides broad non-existence and rigidity theorems for spacelike submanifolds with causal mean curvature, extending to several classes of spacetimes and solving new geometric problems.

Findings

Non-existence of certain spacelike submanifolds in specified spacetimes.

Rigidity results for spacelike submanifolds with causal mean curvature.

Solutions to Calabi-Bernstein and Dirichlet problems in geometric analysis.

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.