A sharp isoperimetric-type inequality for Lorentzian spaces satisfying timelike Ricci lower 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 ${\mathsf{TCD}}_p^e(K,N)$ 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.

Improved both the results and the presentation. 53 pages