#### 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

Email: igor@maths.warwick.ac.uk

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