# Commentarii Mathematici Helvetici

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**Volume 88, Issue 1, 2013, pp. 163–183**

**DOI: 10.4171/CMH/281**

Published online: 2013-01-07

The number of constant mean curvature isometric immersions of a surface

Brian Smyth^{[1]}and Giuseppe Tinaglia

^{[2]}(1) University of Notre Dame, USA

(2) King's College London, UK

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 $x\colon M \to \mathbb R^3$ 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 $\mathbb R^3$, either finite or a circle (Theorem 1.1). When it is a circle then, for the immersion $x$, every cycle in $M$ has vanishing force and, when $H\neq 0$, also vanishing torque. Moreover, we identify closed vector-valued 1-forms whose periods give the force and torque. Our work generalizes a classical result for minimal surfaces to constant mean curvature surfaces.

*Keywords: *Constant mean curvature, minimal surfaces, isometric deformation, rigidity, force, torque

Smyth Brian, Tinaglia Giuseppe: The number of constant mean curvature isometric immersions of a surface. *Comment. Math. Helv.* 88 (2013), 163-183. doi: 10.4171/CMH/281