WEAK CONTINUITY OF THE GAUSS-CODAZZI-RICCI SYSTEM FOR ISOMETRIC EMBEDDING

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 in a fixed coordinate chart is an isometric immersion. 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(loc)(-1)), 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.