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

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.