Weak Continuity of the Gauss-Codazzi-Ricci System for Isometric Embedding

TL;DR

This paper proves the weak continuity of the Gauss-Codazzi-Ricci system for isometric embeddings, establishing a compensated compactness framework that ensures weak limits of approximate solutions are genuine solutions without restrictions on curvature.

Contribution

It introduces a novel compensated compactness framework for the Gauss-Codazzi-Ricci system, allowing weak convergence of approximate solutions to be identified as actual solutions.

Findings

Weak continuity of the Gauss-Codazzi-Ricci system established.

A compensated compactness framework is developed for differential geometry.

Weak limits of approximate solutions are genuine solutions without curvature restrictions.

Abstract

We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform $L^p$-bounded solution sequence for $p>2$, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any sequence of approximate solutions to this system which is uniformly bounded in $L^2$ and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of $H^{-1}_{\text{loc}}$), then the approximating sequence has a weakly convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci system. Furthermore, a minimizing problem is proposed as a selection criterion. For these, no restriction on the Riemann curvature tensor is made.