Lorentzian area and volume estimates for integral mean curvature bounds

TL;DR

This paper derives Lorentzian area and volume estimates for spacetimes under the strong energy condition, linking geometric quantities of initial hypersurfaces to potential convergence results in general relativity.

Contribution

It introduces new Lorentzian geometric estimates based on second fundamental form and mean curvature norms, aiding in convergence analysis of Cauchy developments.

Findings

Established area and volume bounds for spacetimes with strong energy condition

Connected geometric estimates to convergence of initial data sequences

Provided groundwork for future convergence theorems in Lorentzian geometry

Abstract

In the present paper we establish area and volume estimates for spacetimes satisfying the strong energy condition in terms of the area and the $L^n$-norm of the second fundamental form or the mean curvature of an initial Cauchy hypersurface. We believe that these estimates will lay some of the groundwork in establishing new convergence results for Cauchy developments $(M_j, g_j)$ of suitably converging initial data $(\Sigma_j ,h_j ,K_j )$.