Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding

Chen, Gui-Qiang; Slemrod, Marshall; Wang, Dehua

doi:10.1090/S0002-9939-09-10187-9

Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding
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by Gui-Qiang Chen, Marshall Slemrod and Dehua Wang PDF

Proc. Amer. Math. Soc. 138 (2010), 1843-1852 Request permission

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 $H^{-1}_{\text {loc}}$), 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.

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