Bifurcating nodoids

Abstract

All complete, axially symmetric surfaces of constant mean curvature in R^3 lie in the one-parameter family D_tau of Delaunay surfaces. The elements of this family which are embedded are called unduloids; all other elements, which correspond to parameter value tau element in R^-, are immersed and are called nodoids. The unduloids are stable in the sense that the only global constant mean curvature deformations of them are to other elements of this Delaunay family. We prove here that this same property is true for nodoids only when tau is sufficiently close to zero (this corresponds to these surfaces having small `necksizes'). On the other hand, we show that as tau decreases to negative infinity, infinitely many new families of complete, cylindrically bounded constant mean curvature surfaces bifurcate from this Delaunay family. The surfaces in these branches have only a discrete symmetry group.