The number of constant mean curvature isometric immersions of a surface

Details The number of constant mean curvature isometric immersions of a surface The number of constant mean curvature isometric immersions of a surface

2008-11-10

arxiv.org/abs/0811.1231v2

Mathematics

Differential Geometry

21 pages, 1 figure. This paper is now dedicated to Katsumi Nomizu and the references have been updated

Scientific paper

In classical surface theory there are but few known examples of surfaces admitting nontrivial isometric deformations and fewer still non-simply-connected ones. We consider the isometric deformability question for an immersion x: M \to R^3 of an oriented non-simply-connected surface with constant mean curvature H. We prove that the space of all isometric immersions of M with constant mean curvature H is, modulo congruences of R^3, either finite or a circle. When it is a circle then, for the immersion x, every cycle in M has vanishing force and, when H is not 0, also vanishing torque. Our work generalizes a rigidity result for minimal surfaces to constant mean curvature surfaces. Moreover, we identify closed vector-valued 1-forms whose periods give the force and torque.

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