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 $n\geq 4$, 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 M\"uller--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 $n\geq 3$.
Categories: math.DG math.AP