Exact d’Alembertian for Lorentz distance functions
- 1Institute of Mathematics, EPFL, Lausanne, Switzerland
Abstract
We study the distributional 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 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. 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.
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BibTeX
@article{braun2026exact,
title = {{Exact d’Alembertian for Lorentz distance functions}},
author = {Mathias Braun},
journal = {Calculus of Variations and Partial Differential Equations},
year = {2026},
month = sep,
doi = {10.1007/s00526-026-03429-8},
url = {https://doi.org/10.1007/s00526-026-03429-8},
note = {CC BY},
}
RIS
TY - JOUR
TI - Exact d’Alembertian for Lorentz distance functions
AU - Mathias Braun
PY - 2026
DA - 2026/09/01
JO - Calculus of Variations and Partial Differential Equations
DO - 10.1007/s00526-026-03429-8
UR - https://doi.org/10.1007/s00526-026-03429-8
AB - We study the distributional 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 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. 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.
SN - 0944-2669
SN - 1432-0835
ER -