JournalscmhVol. 88, No. 1pp. 163–183

The number of constant mean curvature isometric immersions of a surface

  • Brian Smyth

    University of Notre Dame, USA
  • Giuseppe Tinaglia

    King's College London, UK
The number of constant mean curvature isometric immersions  of a surface cover
Download PDF

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 ⁣:MR3x\colon M \to \mathbb R^3 of an oriented non-simply-connected surface with constant mean curvature HH. We prove that the space of all isometric immersions of MM with constant mean curvature HH is, modulo congruences of R3\mathbb R^3, either finite or a circle (Theorem 1.1). When it is a circle then, for the immersion xx, every cycle in MM has vanishing force and, when H0H\neq 0, 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.

Cite this article

Brian Smyth, Giuseppe Tinaglia, The number of constant mean curvature isometric immersions of a surface. Comment. Math. Helv. 88 (2013), no. 1, pp. 163–183

DOI 10.4171/CMH/281