The threshold theorem for geometric nonlinear wave equations

Abstract

The aim of these notes is to provide a brief overview of a large body recent work whose aim was to prove the Threshold Theorem for energy critical geometric nonlinear wave equations. Within the class of geometric wave equations we include nonlinear wave evolutions which have a geometric structure and origin, including a nontrivial gauge group. The problems discussed here include Wave Maps, Maxwell–Klein–Gordon, as well as the hyperbolic Yang–Mills flow. In a nutshell, the Threshold theorem asserts that these problems are globally well-posed for initial data below the ground state energy.