Wave propagators in the low mass limit

It is a classical result of Duistermaat-Hörmander that on a Lorentzian manifold, any linear hyperbolic differential operator of order 2 (e.g. the d’Alembertian) admits four (4) local distinguished parametrices. These are locally defined approximate fundamental solutions, distinguished by the structure of their singularities, i.e. their wavefront sets.

A central question is whether there are global exact fundamental solutions defined on a complete spacetime, i.e. whether the local parametrices above can be given by global fundamental solutions; the resolution of this question: 1) is highly dependent on global geometric features of the spacetime, 2) requires uniform global phase space analysis, that is, geometric microlocal analysis.

In this talk we present work on this topic for the Klein-Gordon equation with mass m and the wave equation on asymptotically Minkowski spacetimes, where we obtain uniform estimates in the mass down to and including \(m = 0\).

This talk is based on various joint works with Dean Baskin (Texas A&M), Moritz Doll (U. Melbourne), Taira Kouichi (Tokyo University) and Andras Vasy (Stanford)