A Note on Moving Frames along Sobolev Maps and the Regularity of Weakly Harmonic Maps
Luigi Appolloni, Ben Sharp
TL;DR
This work develops a gauge-theoretic framework for moving frames along Sobolev maps and analyzes the regularity of weakly harmonic maps into homogeneous targets. By introducing the Rivière connection $\nabla^{\omega}$ and a corresponding endomorphism $\mathcal R$, the authors show that a small Morrey-norm $\|\omega\|_{M^{2,m-2}}$ yields a finite-energy Coulomb frame that simultaneously trivialises the pulled-back tangent and normal bundles, using a Coulomb gauge reduction and an $\mathcal H^1$–$\mathrm{BMO}$ duality argument to force a gauge $Q$ to be constant. For weakly harmonic maps into homogeneous targets, a small $[u]_{BMO}$ suffices to obtain full regularity, with quantitative bounds on $\|\nabla^2 u\|_{L^1}$ and $\|\nabla u\|_{L^{\infty}}$. The results extend Hélein’s and Rivière’s ideas to higher dimensions and more general homogeneous targets, and include a projection-generalization via $\omega_{\Pi}$, yielding a robust regularity theory under zeroth- or first-order smallness conditions.
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 Rivière 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.