Quantitative Lorentzian isoperimetric inequalities

TL;DR

This paper proves optimal stability estimates for Lorentzian isoperimetric inequalities, linking geometric asymmetry measures to inequality stability, and extends results to Minkowski spacetime with Hausdorff stability.

Contribution

It establishes the first optimal stability estimates for Lorentzian isoperimetric inequalities, refining previous bounds and providing new geometric and Hausdorff stability results.

Findings

Fraenkel asymmetry enters stability quadratically for Bahn--Ehrlich inequality

Linear dependence of asymmetry in Cavalletti--Mondino inequality

Refined inequality recovers quadratic stability behavior

Abstract

We establish optimal stability estimates in terms of the Fraenkel asymmetry with universal dimensional constants for a Lorentzian isoperimetric inequality due to Bahn and Ehrlich and, as a consequence, for a special version of a Lorentzian isoperimetric inequality due to Cavalletti and Mondino. For the Bahn--Ehrlich inequality the Fraenkel asymmetry enters the stability result quadratically like in the Euclidean case while for the Cavalletti--Mondino inequality the Fraenkel asymmetry enters linearly. As it turns out, refining the latter inequality through an additional geometric term allows us to recover the more common quadratic stability behavior. Along the way, we provide simple, self-contained proofs for the above isoperimetric-type inequalities. Moreover, in a fixed conical Minkowski spacetime, we use a Lipschitz bound, naturally provided by the causal structure, to upgrade our quantitative control to a Hausdorff stability estimate. This estimate is formulated in terms of a distance defined by Bahn and Ehrlich, which restricts to a natural Hausdorff-type metric on the space of Cauchy hypersurfaces.