Mathematics

We present a distributional notion of d'Alembertian of a signed time 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 general condition (valid for Finsler spacetimes with curvature bounds), we prove the associated distribution is a signed measure which certifies the integration by parts formula. This expands on the recent elliptic interpretation of the d'Alembertian (e.g. enabling us to give a simplified proof of the splitting theorem) we shortly outline; even in the Lorentzian case, our formulas seem to pioneer its precise meaning across the timelike cut locus. Two central ingredients our work unifies are the localization paradigm of Cavalletti-Mondino and our recent Lorentzian Sobolev calculus. Partly in collaboration with Robert McCann (University of Toronto), Nicola Gigli, Felix Rott (SISSA Trieste), Shin-ichi Ohta (Osaka University), Clemens Sämann (University of Oxford), Argam Ohanyan, Tobias Beran, Matteo Calisti (University of Vienna).