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 of (Mⁿ, g) into the Euclidean space R^n+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. The best result up to date is S Lᵖ and W^2, p for p>n 3 or p=n=2. In this paper, we extend the above result to X whose topology is strictly weaker than W^2, n for n 3. Indeed, X is the weak Morrey space L^p, n-p₂, ₖ 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\`ere--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.