Construction of harmonic coordinates for weak immersions
Dorian Martino, Tristan Rivière
TL;DR
This work establishes a Lorentz–Sobolev harmonic-coordinate theory for weak immersions in even dimensions n≥4. By coupling a Coulomb-frame construction with a continuity argument in the critical Lorentz space L^{(n/2,1)}, it proves the existence of global harmonic coordinates on R^n for induced metrics g_{Φ} when the energy functional E_n(Φ) is small, yielding quantitative proximity to Euclidean geometry in bar{W}^{2,(n/2,1)}. An intrinsic analogue shows harmonic coordinates exist for general metrics with Riem^g ∈ L^{(n/2,1)}, and an extension procedure enables local-to-global Coordinate construction on balls. Applied to immersions, the results yield uniform Sobolev injections under small mean curvature, and they relate the Riemann tensor regularity of the induced metric to the second fundamental form, paving the way for analysis of scale-invariant Willmore-type functionals in higher dimensions. The framework also provides a local version for the unit ball, showing that near-flat graphs admit harmonic-coordinate representations with controlled II-energy.
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 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 $n\geq 3$.