Suppose ℳt 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
are
constant. It is unknown whether there are nontrivial such bendings ℳt. 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 ℳt = ∂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.
Keywords
Isometric embedding, integral mean
curvature, bending, varifolds
