On the Moduli Spaces of Embedded Constant Mean Curvature Surfaces with Three or Four Ends

We are interested in explicitly parametrizing the moduli spaces Mg,k of embedded surfaces in R with finite genus g and a finite number of ends k having constant mean curvature. By rescaling we may assume this constant is 1, the mean curvature of the unit sphere. Two surfaces in R are indentified as points inMg,k if there is isometry of R carrying one surface to the other. Moreover, we shall include inMg,k a somewhat larger class of constant mean curvature (cmc ) surfaces, the Alexandrov embedded surfaces, which are immersed surfaces bounding immersions of handlebodies into R. The structure of these moduli spaces is known: they are finite dimensional real analytic varieties [KMP], but only a few of them are understood completely: the only embedded compact cmc surface is a round sphere [A], so Mg,0 is either a point (g = 0) or empty (g > 0); Mg,1 is empty, since there are no 1-ended examples [M]; and 2-ended examples are necessarily the Delaunay unduloids [KKS], which are simply-periodic surfaces of revolution whose minimal radius or necksize ρ ∈ (0, 1 2 ] parametrizes M0,2, whereasMg,2 is empty for g > 0. The Kapouleas construction [Kp] shows that Mg,k is not empty for every k ≥ 3 and every g. We focus on cmc surfaces with special symmetries: the submoduli space of these can be thought of as the fixed point sets of automorphisms of the (pre)moduli spaces Ng,k of cmc surfaces before modding out by Isom(R). Each embedded end of a cmc surface is asymptotically a Delaunay unduloid [KKS], so we use the necksizes and axes of these asymptotic unduloids to describe a surface and its symmetries. In previous work [G] we considered a subset of the k-unduloids M0,k. We proved existence of an entire connected component of the submoduli space with dihedral symmetry: for k asymptotic axes arranged in a plane with equal angles 2π/k, two k-unduloids exist for (asymptotic) necksizes in the interval (0, 1 k ) and one surface corresponds to the right endpoint. Note that the submoduli space of dihedrally symmetric k-unduloids is one-dimensional.