Exact d'Alembertian for Lorentz distance functions
Mathias Braun
TL;DR
This work develops a comprehensive distributional theory for the $p$-d'Alembertian of signed Lorentz distance functions on metric measure spacetimes under timelike TMCP/TCD bounds. Using Cavalletti–Mondino localization, disintegration along Lorentzian rays, and a Lorentzian Sobolev calculus, it derives exact representation formulas that separate absolutely continuous and singular parts, including contributions from the timelike cut locus, and proves a precise integration-by-parts identity. The authors establish sharp one-sided comparison inequalities for the d'Alembertian and its powers, and extend the theory to a broad, variable-curvature framework with uniqueness results under infinitesimal strict concavity. Applications include a Bochner-type inequality characterizing TCD, synthetic notions of mean curvature and CMC barriers, and Heintze–Karcher-type volume/area bounds that yield volume-singularity theorems in the synthetic Lorentzian setting. The results advance a nonlinear yet elliptic perspective on a fundamentally hyperbolic operator, providing robust tools for synthetic Lorentzian geometry and general relativity in nonsmooth contexts.
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-Sämann. 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.