arXiv: math.DG/9810183

Abstract

Suppose ${\mathcal{ℳ}}_t$ 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

are constant. It is unknown whether there are nontrivial such bendings ${\mathcal{ℳ}}_t$. 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 $S^n$, to wit, if ${\mathcal{ℳ}}_t=\partial X_t$

where $k=-1$ for $H^3$ and $k=1$ for $S^3$. The Euclidean case can be viewed as a special case where $k=0$. 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 a preprint by Rivin and Schlenker. Our result should be compared with the well-known formula of Herglotz.

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