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 $x:M\to {\mathbb{R}}^3$ 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 $H\ne 0$, 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.