On the weak rigidity of isometric immersions of Riemannian and semi-Riemannian manifolds

9 March 2017

Consider a family of uniformly bounded $W^{2,p}$ isometric immersions of an $n$-dimensional (semi-) Riemannian manifold into (resp., semi-) Euclidean spaces. Are the weak limits still isometric immersions?

We answer the question in the affirmative for $p>n$ in the Riemannian case, by exploiting the div-curl structure of the Gauss-Codazzi-Ricci equations, which describe the curvature flatness of the isometric immersions. Along the way a generalised div-curl lemma in Banach spaces is established. Moreover, the endpoint case $p=n=2$ is settled. 

In the semi-Riemannian case we reduce the problem to the weak continuity of H. Cartan's structural equations in $W^{1,p}_{\rm loc}$, which is proved by a generalised compensated compactness theorem relating the weak continuity of quadratic forms to the principal symbols of differential constraints. Again for $p>n$ we obtain the weak rigidity. The case of degenerate hypersurfaces are also discussed, as well as connections to PDEs in fluid dynamics.

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