The number of constant mean curvature isometric immersions of a surface

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.