Wave and Dirac equations on manifolds

TL;DR

This paper reviews recent advances in the analysis of wave and Dirac equations on Lorentzian manifolds, focusing on well-posedness, stability, and geometric structures, especially in black-hole spacetimes like Kerr.

Contribution

It synthesizes recent results on the mathematical properties and structures of hyperbolic equations on Lorentzian manifolds, including index theorems and Green-hyperbolic operators.

Findings

Well-posedness and stability results for initial value problems

Structural analysis of equations on Kerr black-hole spacetimes

Index theorem for hyperbolic Dirac operators

Abstract

We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the index theorem for hyperbolic Dirac operators and properties of the class of Green-hyperbolic operators.