Smyth, Brian; Tinaglia, Giuseppe The number of constant mean curvature isometric immersions of a surface. (English) Zbl 1260.53023 Comment. Math. Helv. 88, No. 1, 163-183 (2013). 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. Cited in 9 Documents MSC: 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature Keywords:constant mean curvature; minimal surfaces; isometric deformation; rigidity; force; torque PDFBibTeX XMLCite \textit{B. Smyth} and \textit{G. Tinaglia}, Comment. Math. Helv. 88, No. 1, 163--183 (2013; Zbl 1260.53023) Full Text: DOI arXiv