{"title":"Nonsmooth d'Alembertian for Lorentz distance functions","authors":"Mathias Braun","doi":"arxiv-2408.16525","DOIUrl":null,"url":null,"abstract":"We refine a recent distributional notion of d'Alembertian of a signed Lorentz\ndistance function to an achronal set in a metric measure spacetime obeying the\ntimelike measure contraction property. We show precise representation formulas\nand comparison estimates (both upper and lower bounds). Under a condition we\ncall \"infinitesimally strict concavity\" (known for infinitesimally Minkowskian\nstructures and established here for Finsler spacetimes), we prove the\nassociated distribution is a signed measure certifying the integration by parts\nformula. This treatment of the d'Alembertian using techniques from metric\ngeometry expands upon its recent elliptic interpretation; even in the smooth\ncase, our formulas seem to pioneer its exact shape across the timelike cut\nlocus. Two central ingredients our contribution unifies are the localization\nparadigm of Cavalletti-Mondino and the Lorentzian Sobolev calculus of Beran et\nal. In the second part of our work, we present several applications of these\ninsights. First, we show the equivalence of the timelike curvature-dimension\ncondition with a Bochner-type inequality. Second, we set up synthetic mean\ncurvature (and barriers for CMC sets) exactly. Third, we prove volume and area\nestimates of Heintze-Karcher-type, which enable us to show several synthetic\nvolume singularity theorems.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.16525","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 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 Lorentzian Sobolev calculus of Beran et
al. 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 (and barriers for CMC sets) exactly. Third, we prove volume and area
estimates of Heintze-Karcher-type, which enable us to show several synthetic
volume singularity theorems.