The number of constant mean curvature isometric immersions of a surface

TL;DR

This paper investigates the space of isometric immersions of non-simply-connected surfaces with constant mean curvature in three-dimensional space, revealing that this space is either finite or forms a circle, thus extending classical rigidity results.

Contribution

It generalizes a known rigidity result from minimal surfaces to constant mean curvature surfaces and characterizes the space of such immersions as finite or circular.

Findings

The space of isometric immersions is either finite or a circle.

When the space is a circle, all cycles have vanishing force; non-zero mean curvature implies vanishing torque.

Identifies closed vector-valued 1-forms related to force and torque periods.

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.