Geometry & Topology Monographs 1 (1998), The Epstein Birthday Schrift, paper no. 1, pages 1-21.

The mean curvature integral is invariant under bending

Frederic J Almgren Jr, Igor Rivin

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.

Keywords. Isometric embedding, integral mean curvature, bending, varifolds

AMS subject classification. Primary: 53A07, 49Q15.

E-print: arXiv:math.DG/9810183

Submitted: 10 May 1998. Published: 21 October 1998.

Notes on file formats

Igor Rivin
Mathematics Institute, University of Warwick
Coventry, CV4 7AL, UK

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