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Suppose
${\mathcal{\mathcal{M}}}_{t}$
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
$${\int}_{{\mathcal{\mathcal{M}}}_{t}}{H}_{t}\phantom{\rule{0.3em}{0ex}}d{\mathcal{\mathscr{H}}}^{2}$$
are constant. It is unknown whether there are nontrivial such bendings
${\mathcal{\mathcal{M}}}_{t}$. 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
${\mathcal{\mathcal{M}}}_{t}=\partial {X}_{t}$
$$d{\int}_{{\mathcal{\mathcal{M}}}_{t}}{H}_{t}\phantom{\rule{0.3em}{0ex}}d{\mathcal{\mathscr{H}}}^{2}=kn1dV\left({X}_{t}\right),$$
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 wellknown results. This,
together with farreaching extensions of the results of the present note is done in a
preprint by Rivin and Schlenker. Our result should be compared with the wellknown
formula of Herglotz.
