Geometric analysis of Lorentzian distance function on spacelike hypersurfaces

TL;DR

This paper investigates the properties of Lorentzian distance functions on spacelike hypersurfaces within curved spacetimes, deriving curvature estimates and conditions for hyperbolicity under specific geometric assumptions.

Contribution

It provides new sharp estimates for the mean curvature of spacelike hypersurfaces and establishes conditions for their hyperbolicity based on Lorentzian distance analysis.

Findings

Sharp mean curvature estimates under curvature bounds

Conditions for hyperbolicity of hypersurfaces

Analysis of Lorentzian distance restricted to hypersurfaces

Abstract

Some analysis on the Lorentzian distance in a spacetime with controlled sectional (or Ricci) curvatures is done. In particular, we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the Omori-Yau maximum principle. As a consequence, and under appropriate hypotheses on the (sectional or Ricci) curvatures of the ambient spacetime, we obtain sharp estimates for the mean curvature of those hypersurfaces. Moreover, we also give a suficient condition for its hyperbolicity.