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The mean curvature integral is invariant under bending

Frederic J Almgren Jr and Igor Rivin

Geometry & Topology Monographs 1 (1998) 1–21

DOI: 10.2140/gtm.1998.1.1

arXiv: math.DG/9810183


Suppose t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals

    H dℋ2
 ℳt  t
are constant. It is unknown whether there are nontrivial such bendings 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 Hn and Sn, to wit, if t = ∂Xt
 ∫        2
d ℳ  Htdℋ  = − kn − 1dV (Xt),
where k = 1 for H3 and k = 1 for S3. 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.


Isometric embedding, integral mean curvature, bending, varifolds

Mathematical Subject Classification

Primary: 49Q15, 53A07


Received: 10 May 1998
Published: 21 October 1998

Frederic J Almgren Jr
Igor Rivin
Mathematics Institute
University of Warwick
United Kingdom