Comparison Theorems in Lorentzian Geometry and applications to spacelike hypersurfaces

TL;DR

This paper develops comparison theorems for Lorentzian distance functions and applies them to estimate curvatures of spacelike hypersurfaces in spacetimes with bounded curvature, advancing geometric analysis in Lorentzian geometry.

Contribution

It introduces Hessian and Laplacian comparison theorems for Lorentzian distance functions using Riccati equation techniques, with applications to curvature estimates of spacelike hypersurfaces.

Findings

Established Hessian and Laplacian comparison theorems for Lorentzian distance functions.

Derived curvature estimates for spacelike hypersurfaces under maximum principles.

Provided new tools for geometric analysis in Lorentzian manifolds.

Abstract

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic operators.