Weak continuity of curvature for connections in L^p

2021/08/30 by Gui‐Qiang Chen, Chen, Gui-Qiang G., Tristan P. Giron +1
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2108.13529

Abstract

We study the weak continuity of two interrelated non-linear partial differential equations, the Yang-Mills equations and the Gauß-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 Gauß-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.

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