Global estimates and energy identities for elliptic systems with antisymmetric potentials

TL;DR

This paper establishes global estimates and energy identities for solutions of elliptic systems with antisymmetric potentials, confirming a conjecture and applying to harmonic maps and surfaces with prescribed mean curvature.

Contribution

It introduces new global estimates and energy identities for elliptic systems with antisymmetric potentials, extending to harmonic maps and confirming Rivi e's conjecture in 2D.

Findings

Global estimates in critical norms for solutions

New energy identities for solution sequences

Confirmation of Rivi e's conjecture in 2D

Abstract

We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover we obtain new energy identities in such norms for sequences of solutions of these systems. The results apply to harmonic maps into general target manifolds and surfaces with prescribed mean curvature. In particular our results confirm a conjecture of Rivi\`ere in the two-dimensional setting.