Decay estimates for Rivière’s equation, with applications to regularity and compactness

Abstract

We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in $\mathbb {R}^2$ introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmonic and almost-harmonic maps from surfaces.