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 S to the Gauss--Codazzi--Ricci equations over a simply-connected closed manifold (Mⁿ,g), one may find an isometric immersion ι of (Mⁿ,g) into the Euclidean space Rⁿ⁺ᵏ whose extrinsic geometry coincides with S. Here the dimension n and the codimension k are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on S and ι. The best result up to date is S ∈ Lᵖ and ι ∈ W2,p for p>n ≥ 3 or $p=n=2$. In this paper, we extend the above result to ι ∈ X whose topology is strictly weaker than W2,n for n ≥ 3. Indeed, X is the weak Morrey space Lp, n-p2,w with arbitrary p ∈ ]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, Rivi\`{e}re--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.