Abstract
All complete, axially symmetric surfaces of constant mean curvature in R3 lie in the one-parameter family Dτ of Delaunay surfaces. The
elements of this family which are embedded are called unduloids; all other
elements, which correspond to parameter value τ ∈ 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 τ is sufficiently close to zero (this corresponds to these surfaces
having small 'necksizes'). On the other hand, we show that as τ decreases to
−∞, 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.
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