$\phi$-parabolicity and the uniqueness of spacelike hypersurfaces immersed in a spatially weighted GRW spacetime

TL;DR

This paper develops new criteria for the parabolicity of complete spacelike hypersurfaces in weighted GRW spacetimes, leading to uniqueness and Calabi-Bernstein type results.

Contribution

It extends existing techniques to weighted GRW spacetimes and establishes new conditions for hypersurface parabolicity and uniqueness.

Findings

Established sufficient conditions for parabolicity of hypersurfaces.

Derived uniqueness results for spacelike hypersurfaces.

Provided Calabi-Bernstein type theorems in this setting.

Abstract

In this paper, we extend a technique due to Romero, Rubio and Salamanca establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker spacetime whose fiber has phi-parabolic universal Riemannian covering. As some applications of this criteria, we obtain uniqueness results concerning spacelikes hypersurfaces immersed in spatially weighted generalized Robertson-Walker spacetimes. Furthermore, Calabi-Bernstein type results are also given.