Exact d'Alembertian for Lorentz distance functions

TL;DR

This paper refines the distributional d'Alembertian for Lorentz distance functions in metric measure spacetimes, providing precise formulas, comparison estimates, and applications to curvature, volume, and singularity theorems.

Contribution

It introduces an exact shape formula for the d'Alembertian across the timelike cut locus and unifies metric geometry techniques with Lorentzian analysis.

Findings

Derived precise representation formulas and bounds for the d'Alembertian.

Proved equivalence of timelike curvature-dimension condition with a Bochner inequality.

Established synthetic volume and area estimates leading to singularity theorems.

Abstract

We refine a recent distributional notion of d'Alembertian of a signed Lorentz distance function to an achronal set in a metric measure spacetime obeying the timelike measure contraction property. We show precise representation formulas and comparison estimates (both upper and lower bounds). Under a condition we call "infinitesimally strict concavity" (known for infinitesimally Minkowskian structures and established here for Finsler spacetimes), we prove the associated distribution is a signed measure certifying the integration by parts formula. This treatment of the d'Alembertian using techniques from metric geometry expands upon its recent nonlinear yet elliptic interpretation; even in the smooth case, our formulas seem to pioneer its exact shape across the timelike cut locus. Two central ingredients our contribution unifies are the localization paradigm of Cavalletti-Mondino and the Sobolev calculus of Beran-Braun-Calisti-Gigli-McCann-Ohanyan-Rott-S\"amann. In the second part of our work, we present several applications of these insights. First, we show the equivalence of the timelike curvature-dimension condition with a Bochner-type inequality. Second, we set up synthetic mean curvature (as well as barriers for CMC sets) exactly. Third, we prove synthetic volume and area estimates of Heintze-Karcher-type, which enable us to show several synthetic volume singularity theorems.