A sharp isoperimetric-type inequality for Lorentzian spaces satisfying timelike Ricci lower bounds

TL;DR

This paper proves a sharp isoperimetric inequality in Lorentzian spaces with timelike Ricci curvature bounds, extending to synthetic spaces and providing bounds relevant for black hole and cosmological spacetime geometries.

Contribution

It introduces a new sharp isoperimetric inequality in Lorentzian pre-length spaces with synthetic Ricci curvature bounds, applicable even in non-smooth settings.

Findings

Establishment of a sharp isoperimetric inequality in Lorentzian spaces.

Application of the inequality to bounds on hypersurface areas in black hole and cosmological spacetimes.

Extension of results to synthetic Lorentzian spaces satisfying timelike Ricci bounds.

Abstract

The paper establishes a sharp and rigid isoperimetric-type inequality in Lorentzian signature under the assumption of Ricci curvature bounded below in the timelike directions. The inequality is proved in the high generality of Lorentzian pre-length spaces satisfying timelike Ricci lower bounds in a synthetic sense via optimal transport, the so-called $\mathsf{TCD}^e_p(K,N)$ spaces. The results are new already for smooth Lorentzian manifolds. Applications include an upper bound on the area of achronal hypersurfaces inside the interior of a black hole (original already in Schwarzschild) and an upper bound on the area of achronal hypersurfaces in cosmological spacetimes.