The number of constant mean curvature isometric immersions of a surface

Brian Smyth; Giuseppe Tinaglia

doi:10.4171/CMH/281

The number of constant mean curvature isometric immersions of a surface

Brian Smyth, Giuseppe Tinaglia

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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.

Original language	English
Pages (from-to)	163-183
Number of pages	21
Journal	COMMENTARII MATHEMATICI HELVETICI
Volume	88
Issue number	1
DOIs	https://doi.org/10.4171/CMH/281
Publication status	Published - 2013

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title = "The number of constant mean curvature isometric immersions of a surface",

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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The number of constant mean curvature isometric immersions of a surface

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