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

Fabio Cavalletti, Andrea Mondino

TL;DR

The paper develops a sharp, rigid isoperimetric-type inequality for Lorentzian spaces under timelike Ricci curvature lower bounds, formulated in the synthetic framework $\,\mathsf{TCD}^{e}_{p}(K,N)$. By localizing along transport rays from a Cauchy hypersurface $V$ and using a Lorentzian Minkowski content, the authors reduce the problem to one-dimensional curvature-dimension estimates and derive a sharp monotonicity formula for the area of $\tau_V$-level sets, culminating in a sharp isoperimetric-type bound for conical equality cases. The main result unifies and extends prior Lorentzian bounds, applies to non-symmetric and low-regularity spacetimes, and yields practical area bounds for spacelike hypersurfaces inside black hole interiors and in cosmological spacetimes. The approach blends synthetic Ricci curvature notions, optimal transport in Lorentzian geodesic spaces, and a new notion of timelike Minkowski content, providing a robust framework for Lorentzian isoperimetric inequalities with rigid equality cases corresponding to conical spacetimes.

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.

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

TL;DR

The paper develops a sharp, rigid isoperimetric-type inequality for Lorentzian spaces under timelike Ricci curvature lower bounds, formulated in the synthetic framework . By localizing along transport rays from a Cauchy hypersurface and using a Lorentzian Minkowski content, the authors reduce the problem to one-dimensional curvature-dimension estimates and derive a sharp monotonicity formula for the area of -level sets, culminating in a sharp isoperimetric-type bound for conical equality cases. The main result unifies and extends prior Lorentzian bounds, applies to non-symmetric and low-regularity spacetimes, and yields practical area bounds for spacelike hypersurfaces inside black hole interiors and in cosmological spacetimes. The approach blends synthetic Ricci curvature notions, optimal transport in Lorentzian geodesic spaces, and a new notion of timelike Minkowski content, providing a robust framework for Lorentzian isoperimetric inequalities with rigid equality cases corresponding to conical spacetimes.

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 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.
Paper Structure (20 sections, 27 theorems, 199 equations)

This paper contains 20 sections, 27 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lorentzian length spaces
  4. Causal Ladder
  5. Optimal transport in Lorentzian geodesic spaces
  6. Geodesics of probability measures in the Lorentz-Wasserstein space
  7. Time-separation functions from sets and their transport relations
  8. Synthetic Timelike Ricci curvature lower bounds
  9. Disintegration of $\mathfrak m$ and regularity of conditional measures
  10. Localization of Timelike Ricci curvature bounds
  11. Localization of timelike Ricci lower bounds to $V$-rays
  12. Timelike Minkowski content and its properties
  13. Properties of timelike Minkowski contents
  14. The unbounded case
  15. The bounded case
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $(M^{n+1},g)$ be a globally hyperbolic Lorentzian manifold satisfying Hawking-Penrose's strong energy condition (i.e., ${\rm Ric}\geq 0$ on timelike vectors) and let $V \subset M$ be Cauchy hypersurface. Then for any compact and acausal hypersurface $S\subset I^+(V)$ the following inequality is

Theorems & Definitions (65)

  • Example 1.1
  • Theorem 1.2: A sharp isoperimetric-type inequality
  • Remark 1.3
  • Theorem 1.4: Area Monotonicity
  • Remark 1.5
  • Definition 2.1: Causal space, Kronheimer-Penrose CausalSpace
  • Definition 2.2: Lorentzian pre-length space $(X,\mathsf d, \ll, \leq, \tau)$
  • Definition 2.3: Causal/timelike curves
  • Definition 2.4: Timelike non-branching
  • Proposition 2.5
  • ...and 55 more