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

TL;DR

This paper proves that low-regularity $L^p$-connections with prescribed curvature can be approximated by smooth ones and applies this to establish existence and characterization results for low-regularity isometric immersions.

Contribution

It introduces an elementary approximation method for $L^p$-connections with small norm and applies it to $W^{2,p}$-isometric immersions, clarifying global versus local issues.

Findings

$L^p$-connections with small norm can be approximated by smooth connections with the same curvature

Existence of $W^{2,p}$-isometric immersions is established for weak solutions of Gauss--Codazzi--Ricci equations

Characterization of metrics admitting $W^{2,p}$ but not smooth isometric immersions

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 $\Omega$ is an $L^p$-connection $1$-form on a vector bundle over a closed Riemannian $n$-manifold $\mathcal{M}$ with small $L^p$-norm ($p>n$) and smooth curvature $2$-form $\mathscr{F}$, then $\Omega$ 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 $\mathcal{M}^n \to \mathbf{R}^{n+k}$ 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 $\mathcal{M}$ that admit $W^{2,p}$- but no $C^\infty$-isometric immersions.