A Note on Moving Frames along Sobolev Maps and the Regularity of Weakly Harmonic Maps

TL;DR

This paper establishes conditions under which weakly differentiable maps between Riemannian manifolds have trivialized tangent bundles and are fully regular, using Morrey-norm and BMO seminorm smallness conditions, with implications for weakly harmonic maps.

Contribution

It introduces new structure equations and combines Coulomb-frame methods with Hardy-BMO duality to prove regularity and trivialization results for weakly harmonic maps.

Findings

Small Morrey-norm of the gradient ensures trivialized tangent bundle.

Small BMO seminorm of the map guarantees full regularity.

Results apply to maps between manifolds of any dimension and into homogeneous targets.

Abstract

The purpose of this note is twofold. First we show that, for weakly differentiable maps between Riemannian manifolds of any dimension, a smallness condition on a Morrey-norm of the gradient is sufficient to guarantee that the pulled-back tangent bundle is trivialised by a finite-energy frame over simply connected regions in the domain. This is achieved via new structure equations for a connection introduced by Rivi\`ere in the study of weakly harmonic maps, combined with Coulomb-frame methods and the Hardy-BMO duality of Fefferman-Stein. We also prove that for weakly harmonic maps from domains of any dimension into closed homogeneous targets, a smallness condition on the BMO seminorm of the map is sufficient to obtain full regularity.