# The number of constant mean curvature isometric immersions of a surface

### Brian Smyth

University of Notre Dame, USA### Giuseppe Tinaglia

King's College London, UK

## 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 $x\colon M \to \mathbb R^3$ 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.

## Cite this article

Brian Smyth, Giuseppe Tinaglia, The number of constant mean curvature isometric immersions of a surface. Comment. Math. Helv. 88 (2013), no. 1, pp. 163–183

DOI 10.4171/CMH/281