What question did this study set out to answer?

To explore the consequences of smallness conditions on Morrey and BMO norms for weakly harmonic maps and Sobolev maps.

January 16, 2026Proceedings of the American Mathematical Society0 citations

A note on moving frames along Sobolev maps and the regularity of weakly harmonic maps

LALuigi AppolloniUniversity of Leeds BSBen SharpUniversity of Leeds

Key Points

To explore the consequences of smallness conditions on Morrey and BMO norms for weakly harmonic maps and Sobolev maps.
Analyzed weakly differentiable maps between Riemannian manifolds.
Utilized new structure equations for connections and Coulomb-frame methods.
Applied Hardy-BMO duality of Fefferman-Stein for deriving results.
Smallness condition on the Morrey-norm of the gradient guarantees trivialisation of the pulled-back tangent bundle in simply connected regions.
A smallness condition on the BMO seminorm ensures full regularity for weakly harmonic maps from any domain to closed 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 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.

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