#### Volume 1 (1998)

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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
##### Abstract
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Suppose ${\mathsc{ℳ}}_{t}$ 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

${\int }_{{\mathsc{ℳ}}_{t}}{H}_{t}\phantom{\rule{0.3em}{0ex}}d{\mathsc{ℋ}}^{2}$

are constant. It is unknown whether there are nontrivial such bendings ${\mathsc{ℳ}}_{t}$. 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 ${\mathsc{ℳ}}_{t}=\partial {X}_{t}$

$d{\int }_{{\mathsc{ℳ}}_{t}}{H}_{t}\phantom{\rule{0.3em}{0ex}}d{\mathsc{ℋ}}^{2}=-kn-1dV\left({X}_{t}\right),$

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.

##### Keywords
Isometric embedding, integral mean curvature, bending, varifolds
##### Mathematical Subject Classification
Primary: 53A07, 49Q15