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 $n\geq 4$ 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 any even dimension $n\geq 4$ of a famous result of Müller–Šverák [36] for $n=2$. The existence of such coordinates is a key tool used by the authors in [34] 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 $n\geq 3$.