The mean curvature integral is invariant under bending

Abstract

Suppose M_t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E^3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals over M_t are constant. It is unknown whether there are nontrivial such bendings. The estimates also hold for periodic manifolds for which there are nontrivial bendings. In addition, our methods work essentially without change to show the similar results for submanifolds of H^n and S^n. The rigidity of the mean curvature integral can be used to show new rigidity results for isometric embeddings and provide new proofs of some well-known results. This, together with far-reaching extensions of the results of the present note is done in the preprint: I Rivin, J-M Schlenker, Schlafli formula and Einstein manifolds, IHES preprint (1998). Our result should be compared with the well-known formula of Herglotz.