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arXiv · 2401.03949

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

Fabio Cavalletti·Andrea Mondino
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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.

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Fabio Cavalletti, Andrea Mondino. 2024-01-08. A sharp isoperimetric-type inequality for Lorentzian spaces satisfying timelike Ricci lower bounds. https://arxiv.org/abs/2401.03949

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