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The number of constant mean curvature isometric immersions of a surface

Research output: Contribution to journalArticle

Brian Smyth, Giuseppe Tinaglia

Original languageEnglish
Pages (from-to)163-183
Number of pages21
JournalCOMMENTARII MATHEMATICI HELVETICI
Volume88
Issue number1
DOIs
Published2013

Bibliographical note

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

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    Uploaded date:21 Jul 2015

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Abstract

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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