Smoothability of L^p-connections on bundles and isometric immersions with W^2,p-regularity

2020/10/26 by Siran Li, Li, Siran
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:35F50 #msc:53C07

paper · pdf · doi:10.48550/arxiv.2010.13560

This note replaces an earlier version, ArXiv Preprint: 2003.05595

arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

We are concerned with two interrelated problems: smoothability of connection 1-forms with low regularity on bundles with prescribed smooth curvature 2-forms, and existence of isometric immersions with low regularity. We first show that if Ω is an L^p-connection 1-form on a vector bundle over a closed Riemannian n-manifold M with small L^p-norm (p>n) and smooth curvature 2-form \mathscrF, then Ω can be approximated in the L^p_{\rm loc}-topology by smooth connections of the same curvature (not necessarily gauge equivalent). Our proof, adapted from S. Mardare's work on the fundamental theory of surfaces with L^p-second fundamental form, is elementary in nature and uses only Hodge decomposition and fixed point theorems. This result is then applied to the study of isometric immersions of Riemannian manifolds with low regularity. We revisit the proof for the existence of W^2,p-isometric immersion Mⁿ → R^n+k with arbitrary n and k given weak solutions to the Gauss--Codazzi--Ricci equations, aiming at elucidating some global vs. local issues, and also we provide a characterisation for metrics on M that admit W^2,p- but no C^∞-isometric immersions.

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