Embedded constant mean curvature surfaces in Euclidean three-space

In this paper we refine the construction and related estimates for complete constant mean curvature surfaces in Euclidean three-space developed in Kapouleas (Ann Math 131:239–330, 1990 ) by adopting the more precise and powerful version of the methodology which was developed in Kapouleas (Invent Math 119(3):443–518, 1995 ). As a consequence we remove the severe restrictions in establishing embeddedness for complete Constant Mean Curvature surfaces in Kapouleas (Ann Math 131:239–330, 1990 ) and we produce a very large class of new embedded examples of finite topology.