On the fundamental theorem of submanifold theory and isometric immersions with supercritical low regularity

A fundamental result in global analysis and nonlinear elasticity asserts that given a solution $\mathfrak{S}$ to the Gauss–Codazzi–Ricci equations over a simply-connected closed manifold $({\mathcal{M}}^n,g)$, one may find an isometric immersion $\iota$ of $({\mathcal{M}}^n,g)$ into the Euclidean space ${\mathbb{R}}^{n+k}$ whose extrinsic geometry coincides with $\mathfrak{S}$. Here the dimension $n$ and the codimension $k$ are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on $\mathfrak{S}$ and $\iota$. The best result up to date is $\mathfrak{S}\in L^p$ and $\iota \in W^{2,p}$ for $p>n\ge 3$ or $p=n=2$. In this paper, we extend the above result to $\iota \in \mathcal{X}$ whose topology is strictly weaker than $W^{2,n}$ for $n\ge 3$. Indeed, $\mathcal{X}$ is the weak Morrey space $L_{2,w}^{p,n-p}$ with arbitrary $p\in ]2,n]$. This appears to be first supercritical result in the literature on the existence of isometric immersions with low regularity, given the solubility of the Gauss–Codazzi–Ricci equations. Our proof essentially utilises the theory of Uhlenbeck gauges – in particular, Rivière–Struwe’s work [Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 (2008)] on harmonic maps in arbitrary dimensions and codimensions – and compensated compactness.