On the Geometry of Conformally Stationary Lorentz Spaces

TL;DR

This paper explores the geometry of conformally stationary Lorentz manifolds, extending classical results and establishing new Bernstein-type theorems and stability properties for spacelike hypersurfaces within these spaces.

Contribution

It extends classical geometric results to conformally stationary Lorentz spaces, including Bernstein-type theorems and stability analyses for hypersurfaces.

Findings

Extended Simons' result to conformally stationary Lorentz spaces.

Constructed complete, noncompact maximal submanifolds in de Sitter and anti-de Sitter spaces.

Proved Bernstein-type theorems without constant mean curvature assumption.

Abstract

In this paper we study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending to these spaces a result of J. Simons on the minimality of cones in Euclidean space, and apply it to the construction of complete, noncompact maximal submanifolds of both de Sitter and anti-de Sitter spaces. Then we state and prove very general Bernstein-type theorems for spacelike hypersurfaces in conformally stationary Lorentz manifolds, one of which not assuming the hypersurface to be of constant mean curvature. Finally, we study the strong $r$-stability of spacelike hypersurfaces of constant $r$-th mean curvature in a conformally stationary Lorentz manifold of constant sectional curvature, extending previous results in the current literature.