Spacelike Foliations on Lorentz manifolds

TL;DR

This paper explores the geometric characteristics of spacelike foliations on Lorentz manifolds, establishing conditions for stability, umbilicity, and geodesicity of the leaves, especially under constant mean curvature and compactness assumptions.

Contribution

It provides new conditions under which spacelike leaves are stable, totally umbilic, or totally geodesic in Lorentz manifolds with a timelike conformal vector field.

Findings

Stable leaves under constant mean curvature.

Compact spacelike hypersurfaces are totally umbilic.

Noncompact hypersurfaces are totally geodesic using Maximum Principle.

Abstract

In this work, we study the geometric properties of spacelike foliations by hypersurfaces on a Lorentz manifold. We investigate conditions for the leaves being stable, totally geodesic or totally umbilical. We consider that $\overline{M}^{n+1}$ is equipped with a timelike closed conformal vector field $\xi$. If the foliation has constant mean curvature, we show that the leaves are stable. When the leaves are compact spacelike hypersurfaces we show that, under certain conditions, its are totally umbilic hypersurfaces. In the case of foliations by complete noncompact hypersurfaces, we using a Maximum Principle at infinity to conclude that the foliation is totally geodesic.