Classification of Embedded Constant Mean Curvature Surfaces with Genus Zero and Three Ends

For each embedded constant mean curvature surface in R 3 with three ends and genus zero, we construct a conjugate cousin boundary contour in S 3. The moduli space of such contours is parametrized by the space of triples of distinct points in S 2. This imposes necessary conditions on the cmc surfaces; moreover, we expect the space of triples exactly parametrizes their moduli space. Our approach extends to those cmc surfaces with k ends and genus 0 which have a reeection, and suggests properties of the higher genus case. Surfaces of constant mean curvature H in R 3 arise naturally from minimization of surface area with a volume constraint. If H 0, we have a minimal surface; otherwise, by rescaling we may assume that H 1, and we call this a cmc surface. We are interested in classifying complete embedded cmc surfaces and in parametrizing their moduli spaces. Embeddedness is not only physically natural, but also seems necessary to give a tractable problem. For example, Alexandrov showed the unit sphere is the only embedded compact cmc surface A], whereas Wente constructed nonembedded compact examples W]; many more are now known, especially by the work of Kapouleas Kp1{3] and by an integrable systems approach PS, Bo, EKT] which gives multidimensional families of examples. The constructions of Kapouleas also suggest that arbitrary cmc surfaces can be glued together in much more general ways (for a survey, see MP]), showing the moduli spaces of immersed cmc surfaces must be very rich. We will consider the slightly more general class of almost embedded cmc surfaces, which are those that bound an immersed three-manifold. These are the surfaces to which the Alexandrov reeection technique can be applied A, KKS]; they share the basic structure of embedded cmc surfaces, but the interior three-manifold is allowed to overlap itself. It seems that this class is most natural mathematically, though it is diicult to determine a priori exactly which of these are actually embedded. Furthermore, we want to restrict our attention to cmc surfaces of nite toplogy (excluding , for instance, triply periodic surfaces). A fundamental result KKS] shows that each end of such a surface is asymptotically an unduloid, an embedded cmc surface of revolution found by Delaunay D]. This leads us to call any almost embedded cmc surface with genus g and k ends a k-unduloid (of genus g), or triunduloid when k = 3.