Weak continuity of curvature for connections in $L^p$

TL;DR

This paper demonstrates that certain geometric PDEs involving $L^p$-connections, like Yang-Mills and Gau{a}ss-Codazzi-Ricci equations, exhibit weak continuity due to inherent cancellations, enabling passage to limits without dimension restrictions or gauge transformations.

Contribution

It establishes the weak continuity of Yang-Mills and Gau{a}ss-Codazzi-Ricci equations for $L^p$-connections, highlighting the role of div-curl structures and cancellations.

Findings

Weak convergence of Yang-Mills connections preserves solutions.

Weak limits of isometric immersions maintain curvature equations.

Methods are independent of dimension and gauge choices.

Abstract

We study the weak continuity of two interrelated non-linear partial differential equations, the Yang-Mills equations and the Gau{\ss}-Codazzi-Ricci equations, involving $L^p$-integrable connections. Our key finding is that underlying cancellations in the curvature form, especially the div-curl structure inherent in both equations, are sufficient to pass to the limit in the non-linear terms. We first establish the weak continuity of Yang-Mills equations and prove that any weakly converging sequence of weak Yang-Mills connections in $L^p$ converges to a weak Yang-Mills connection. We then prove that, for a sequence of isometric immersions with uniformly bounded second fundamental forms in $L^p$, the curvatures are weakly continuous, which leads to the weak continuity of the Gau{\ss}-Codazzi-Ricci equations with respect to sequences of isometric immersions with uniformly bounded second fundamental forms in $L^p$. Our methods are independent of dimensions and do not rely on gauge changes.