Commentarii Mathematici Helvetici
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Published online: 2013-01-07
The number of constant mean curvature isometric immersions of a surfaceBrian Smyth and Giuseppe Tinaglia (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