The First Bifurcation Point for Delaunay Nodoids

are constant mean curvature (CMC) surfaces of revolution, and they are translationally periodic. By a rigid motion and homothety of R we may place the Delaunay surfaces so that their axis of revolution is the x1-axis and their constant mean curvature is H = 1 (henceforth we assume this). We consider the profile curve in the half-plane {(x1, 0, x3) ∈ R |x3 > 0} that gets rotated about the x1-axis to trace out a Delaunay surface. This curve alternates periodically between maximal and minimal heights (with respect to the positive x3 direction), which we refer to as the bulge radius and the neck radius, respectively, of the Delaunay surface. Let us denote the neck radius by r. Delaunay surfaces come in two 1-parameter families: one is a family of embedded surfaces called unduloids that can be parametrized by the neck radius r ∈ (0, 1/2]; the other is a family of nonembedded surfaces called nodoids that can be parametrized by the neck radius r ∈ (0,∞]. For unduloids, r = 1/2 gives the round cylinder. For both unduloids and nodoids, the limiting degenerate surface as r → 0 is a chain of tangent spheres of radii 1 centered along the x1-axis. In this paper we shall be concerned with nodoids. We will see that a common bifurcation point for Delaunay nodoids is encountered in the following two distinctly different approaches for constructing CMC surfaces: (1) Using analytic techniques, Mazzeo and Pacard [12] showed existence of a finite value r0 so that for neck radii r < r0 the nodoids are nonbifurcating, and for r > r0 the nodoids can bifurcate. They showed that bifurcating nodoids deform smoothly through families of CMC surfaces that are of bounded distance from a fixed line yet are not surfaces of revolution. They also gave an analytic gluing construction for CMC surfaces with asymptotically Delaunay ends (i.e. each end converges to an end of a Delaunay surface) that works only when each end converges to a nonbifurcating Delaunay end, that is, each end converges either to an unduloid, or to a nodoid with neck size less than r0. Their gluing construction involves adding ends of neck radii r close to zero to a preexisting CMC surface. Of course those r close to zero are less than r0, but the construction also requires that all the asymptotically nodoid ends of the preexisting surface satisfy r < r0. So the surfaces constructed will necessarily have asymptotic neck radii r < r0 at all asymptotically nodoidal ends. There are a number of other gluing constructions along these lines for making CMC surfaces, and they always involve the Delaunay ends being nonbifurcating. (See the works of Kusner, Mazzeo, Pacard, Pollack, Ratzkin [11], [13], [14], [17] for more on this.) Mazzeo and Pacard gave a clear reason for the existence of this bifurcation point r0, in terms of the existence of nontrivial nullity for a particular Jacobi operator associated to the second variation formula for Delaunay surfaces, but they did not compute the precise value of r0. (2) Using integrable systems techniques developed by Dorfmeister, Pedit andWu in [7], Dorfmeister, Wu [8] and Schmitt [19] (see also [9]) constructed genus 0 CMC surfaces with three asymptotically Delaunay ends. In [8] the construction was restricted to surfaces with asymptotically unduloidal ends, because such ends are embedded. However, the construction in [19] and [9] includes asymptotically nodoidal ends as well. The construction begins with the selection of