2008/11/10 by Brian Smyth, Smyth, Brian, Giuseppe Tinaglia +1
Mathematics · #53A10 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53A10
paper · pdf · doi:10.48550/arxiv.0811.1231
21 pages, 1 figure. This paper is now dedicated to Katsumi Nomizu and the references have been updated
arxiv created 2008/11/14 · arxiv updated 2009/12/01
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 → R3 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 R3, 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.