Construction of harmonic coordinates for weak immersions

Abstract

We prove that any weak immersion in the critical Sobolev space $W^{\frac{n}{2}+1,2}({\mathbb{R}}^n;{\mathbb{R}}^d)$ in even dimension $n\ge 4$, has global harmonic coordinates if its second fundamental form is small in the Sobolev space $W^{\frac{n}{2}-1,2}({\mathbb{R}}^n;{\mathbb{R}}^d)$. This is a generalization to arbitrary even dimension $n\ge 4$ of a famous result of Müller--Sverak \cite{muller1995} for $n=2$. The existence of such coordinates is a key tool used by the authors in \cite{MarRiv20252} for the analysis of scale-invariant Lagrangians of immersions, such as the Graham--Reichert functional. From a purely intrinsic perspective, the proof of the main result leads to a general local existence theorem of harmonic coordinates for general metrics with Riemann tensor in $L^p$ for any $p>n/2$ in any dimension $n\ge 3$.