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

Abstract

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.

This paper corrects and supercedes 2003.05595