A characterization of compact surfaces with constant mean curvature

Umehara, Masaaki

doi:10.1090/S0002-9939-1990-0987616-2

A characterization of compact surfaces with constant mean curvature
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Proc. Amer. Math. Soc. 108 (1990), 483-489 Request permission

Abstract:

Surfaces in a $3$-space of constant curvature, whose arbitrary sufficiently small open subsets admit a non-trivial isometric deformation preserving the mean curvature function, are called locally $H$-deformable. It is well known that surfaces with constant mean curvature which are not totally umbilical are all locally $H$-deformable. Conversely, we shall show in this paper that any compact locally $H$-deformable surface has constant mean curvature.

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