Geometric Wave Equations

TL;DR

This paper provides a comprehensive overview of the solution theory for geometric wave equations in Lorentzian geometry, focusing on Green functions, the Cauchy problem, and Poisson algebras, with detailed proofs and distribution theory background.

Contribution

It offers a detailed treatment of existence, uniqueness, and solution methods for geometric wave equations on globally hyperbolic manifolds, including new insights into Poisson algebra relations.

Findings

Existence and uniqueness of Green functions for hyperbolic operators

Solution construction for the Cauchy problem on globally hyperbolic manifolds

Connection between initial value Poisson algebra and dynamical Poisson algebra

Abstract

In these lecture notes we discuss the solution theory of geometric wave equations as they arise in Lorentzian geometry: for a normally hyperbolic differential operator the existence and uniqueness properties of Green functions and Green operators is discussed including a detailed treatment of the Cauchy problem on a globally hyperbolic manifold both for the smooth and finite order setting. As application, the classical Poisson algebra of polynomial functions on the initial values and the dynamical Poisson algebra coming from the wave equation are related. The text contains an introduction to the theory of distributions on manifolds as well as detailed proofs.