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 of an oriented non-simply-connected surface with constant mean curvature . We prove that the space of all isometric immersions of with constant mean curvature is, modulo congruences of , either finite or a circle (Theorem 1.1). When it is a circle then, for the immersion , every cycle in has vanishing force and, when , 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.
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Brian Smyth, Giuseppe Tinaglia, The number of constant mean curvature isometric immersions of a surface. Comment. Math. Helv. 88 (2013), no. 1, pp. 163–183DOI 10.4171/CMH/281