Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces

TL;DR

This paper proves that wave maps from 2D Minkowski space to hyperbolic spaces are globally smooth for smooth initial data, assuming certain local theory claims and analyzing potential special solutions to rule out singularities.

Contribution

It introduces a strategy based on almost periodic solutions and stress-energy tensor analysis to establish global regularity for wave maps with large energy.

Findings

Reduction to almost periodic solutions in the energy class

Analysis of stress-energy tensor to identify special solutions

Framework for ruling out self-similar and traveling solutions

Abstract

We show that wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$ are globally smooth in time if the initial data is smooth, conditionally on some reasonable claims concerning the local theory of such wave maps, as well as the self-similar and travelling (or stationary solutions); we will address these claims in the sequels \cite{tao:heatwave2}, \cite{tao:heatwave3}, \cite{tao:heatwave4} to this paper. Following recent work in critical dispersive equations, the strategy is to reduce matters to the study of an \emph{almost periodic} maximal Cauchy development in the energy class. We then repeatedly analyse the stress-energy tensor of this development (as in \cite{tao:forges}) to extract either a self-similar, travelling, or degenerate non-trivial energy class solution to the wave maps equation. We will then rule out such solutions in the sequels to this paper, establishing the desired global regularity result for wave maps.