A Remark on Gauge Transformations and the Moving Frame Method

TL;DR

This paper provides a concise proof of existing regularity results in geometric analysis by employing the moving frame method and elementary calculus of variations, simplifying previous approaches without introducing new results.

Contribution

It offers a shorter, more elementary proof of regularity results using the moving frame method, avoiding complex theorems like Nash-Moser.

Findings

Simplified proof of regularity results

Use of elementary calculus of variations

Potential to avoid Nash-Moser theorem in certain proofs

Abstract

In this note we give a shorter proof of recent regularity results by Riviere and Riviere-Struwe. We differ from the mentioned articles only in using the direct method of Helein's moving frame to construct a suitable gauge transformation. Though this is neither new nor surprising, it enables us to describe a proof of regularity using besides the duality of Hardy- and BMO-space only elementary arguments of calculus of variations and algebraic identities. Moreover, we remark that in order to prove Hildebrandt's conjecture one can avoid the Nash-Moser imbedding theorem. There are no new results presented here, nor are there any techniques we could claim originality for.