The number of constant mean curvature isometric immersions of a surface
Loading...
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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.
21 pages, 1 figure. This paper is now dedicated to Katsumi Nomizu and the references have been updated
21 pages, 1 figure. This paper is now dedicated to Katsumi Nomizu and the references have been updated