The number of constant mean curvature isometric immersions of a surface (Q1939566)

Summary: 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 \(H\neq 0\), also vanishing torque. Moreover, we identify closed vector-valued one-forms whose periods give the force and torque. Our work generalizes a classical result for minimal surfaces to constant mean curvature surfaces.