An elliptic perspective on the d'Alembert operator from general relativity (Geometry Seminar)

We present a distributional notion of d'Alembertian of Lorentz distance functions in spacetimes which may be smooth or nonsmooth. We derive comparison estimates and precise representation formulas. This expands on a recent elliptic interpretation of this operator (e.g. enabling us to give a simplified proof of the classical Lorentzian splitting theorem); even in the Lorentzian case, our results seem to pioneer its interpretation 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), Clemens Sämann (University of Oxford), Argam Ohanyan, Tobias Beran, Matteo Calisti (University of Vienna).