Moduli Spaces of Embedded Constant Mean Curvature Surfaces with Few Ends and Special Symmetry

We give necessary conditions on complete embedded cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are twodimensional varieties in the moduli spaces of general cmc surfaces. We characterize fundamental domains of our cmc surfaces by associated great circle polygons in the three-sphere. We are interested in explicitly parametrizing the moduli space Mg,k of complete, connected, properly embedded surfaces in R with finite genus g and a finite number k of labeled ends k having nonzero constant mean curvature. By rescaling we may assume this constant is 1, the mean curvature of the unit sphere. Two surfaces in R are identified as points in Mg,k if there is a rigid motion of R carrying one surface to the other. Moreover, we shall include in Mg,k a somewhat larger class of constant mean curvature (cmc) surfaces, the almost (or Alexandrov) embedded surfaces, which are immersed surfaces bounding immersions of handlebodies into R. The space Mg,k is a finite dimensional real analytic variety, and in a neighborhood of a surface with no L-Jacobi fields it is a (3k − 6)-dimensional manifold [KMP] for each k ≥ 3. A few of the cmc moduli spaces Mg,k are known explicitly: 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 one-ended examples [M]; two-ended examples are necessarily the Delaunay unduloids [KKS], which are simply-periodic surfaces of revolution whose minimal radius or neckradius ρ ∈ (0, 1 2 ] parametrizes M0,2, while Mg,2 is empty for g > 0. The Kapouleas construction [Kp] shows that Mg,k is not empty for every k ≥ 3 and every g. Furthermore, an embedded end of a cmc surface is asymptotically a Delaunay unduloid [KKS], and this defines in particular the neckradii and axes of the ends, which are related via a balancing formula. We call the surfaces in Mg,k the k-unduloids of genus g, or simply k-unduloids if their genus is 0. In the present paper we focus on cmc surfaces with few ends and special symmetries: these give two-dimensional submoduli spaces ofMg,3 andMg,4. The triunduloids of genus g comprising Mg,3 are special in that a priori each has a plane of reflection symmetry [KKS], which we will think of as a horizontal plane. Their moduli space is 3-dimensional at nondegenerate points [KMP]. Here we study the Y-shaped isosceles triunduloids inM0,3 which Date: October 1996. Supported by SFB 256 at Universität Bonn, and NSF grant DMS 94-04278 at UMassAmherst. 1 2 GROSSE-BRAUCKMANN AND KUSNER