2024/05/25 by Siran Li, Li, Siran, Xiangxiang Su +1
Materials Science · Mathematics · Medicine · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Muscle and Compartmental Disorders #Polymer Foaming and Composites
paper · pdf · doi:10.48550/arxiv.2405.16249
openalex publication_date 2024/05/25 · openalex created_date 2024/05/29 · openalex updated_date 2026/07/28
A fundamental result in global analysis and nonlinear elasticity asserts that given a solution \mathfrakS 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 \mathfrakS. Here the dimension n and the codimension k are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on \mathfrakS and ι. The best result up to date is \mathfrakS ∈ 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, 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.