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Quantitative Lorentzian isoperimetric inequalities

Christian Lange, Jonas W. Peteranderl

TL;DR

The paper proves sharp, quantitative stability bounds for Lorentzian isoperimetric inequalities, specifically the Bahn–Ehrlich and Cavalletti–Mondino inequalities, by expressing achronal Lipschitz hypersurfaces as graphs over the hyperboloid and deriving explicit $L^1$-type deficits. It shows that the BE deficit controls the Fraenkel asymmetry with quadratic dependence, while the CM deficit yields linear dependence, and a refined CM formulation recovers quadratic stability; all proofs are self-contained and provide sharpness. The approach blends geometric and functional-analytic methods, including a quantitative Minkowski inequality and compactness reductions, yielding transparent, explicit constants. The results illuminate stability phenomena in Lorentzian geometry with potential applications to relativistic geometry and spacetime singularity contexts, and establish a clear link between Lorentzian and Euclidean stability paradigms via carefully designed deficits and asymmetries.

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.

Quantitative Lorentzian isoperimetric inequalities

TL;DR

The paper proves sharp, quantitative stability bounds for Lorentzian isoperimetric inequalities, specifically the Bahn–Ehrlich and Cavalletti–Mondino inequalities, by expressing achronal Lipschitz hypersurfaces as graphs over the hyperboloid and deriving explicit -type deficits. It shows that the BE deficit controls the Fraenkel asymmetry with quadratic dependence, while the CM deficit yields linear dependence, and a refined CM formulation recovers quadratic stability; all proofs are self-contained and provide sharpness. The approach blends geometric and functional-analytic methods, including a quantitative Minkowski inequality and compactness reductions, yielding transparent, explicit constants. The results illuminate stability phenomena in Lorentzian geometry with potential applications to relativistic geometry and spacetime singularity contexts, and establish a clear link between Lorentzian and Euclidean stability paradigms via carefully designed deficits and asymmetries.

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.

Paper Structure

This paper contains 20 sections, 11 theorems, 79 equations.

Table of Contents

  1. Introduction and main results
  2. Comparison with the stability of functional inequalities
  3. Structure of the paper
  4. Preliminaries
  5. Some basic Lorentzian geometry
  6. Conical Minkowski spacetimes and Cauchy hypersurfaces
  7. Induced length metrics
  8. Functional representations and Bahn--Ehrlich isoperimetric inequality
  9. Timelike curves
  10. Graphs and functional representations
  11. Two functional-analytic proofs of the Bahn--Ehrlich inequality
  12. Stability of the Bahn--Ehrlich isoperimetric inequality
  13. A geometric proof of the Bahn--Ehrlich inequality
  14. A quantitative Minkowski inequality
  15. Notion of distance
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $M$ be a conical Minkowski spacetime in $\mathbb{L}^{n+1}$, $n\in \mathbb{N}$. Every achronal Lipschitz hypersurface $S$ of $M$ with $V(C(S))<\infty$ and $\mu(\pi(S))<\infty$ satisfies Moreover, the exponent $2$ of the Fraenkel asymmetry is optimal.

Theorems & Definitions (21)

  • Theorem 1.1: Stability for a generalized Bahn--Ehrlich isoperimetric inequality
  • Proposition 1.2: Relation between isoperimetric deficits
  • Corollary 1.3: Stability for the CM-inequality \ref{['eq:CM_ineq']}
  • Corollary 1.4: Stability for the refined CM-inequality \ref{['eq:deltaCMstar']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof : First proof of Theorem \ref{['thm:Bahn_Ehrlich_gen']}
  • ...and 11 more