Constant Mean Curvature  (CMC) surfaces  are critical points for  the area functional subject to an enclosed volume constraint. The classical  examples of finite topological type are  round spheres and cylinders, as well a family of rotationally invariant surfaces discovered by Delaunay in 1841. Kapouleas and others developed a gluing methodology which produced embedded examples of many topological types. 

 

The central idea of these constructions is to produce families of surfaces that serve as approximate solutions, and to find an exact solution as a normal perturbation of one of the approximate solutions.  These constructions tend to be extremely technical, stemming from the fact that families of approximate solutions are divergent  in the moduli space. We present a construction based upon an alternative gluing technique introduced by   Mazzeo and Picard, which simplifies many of the most technical points. Roughly speaking, we construct surfaces from several  standard pieces by matching Cauchy data. Our work further develops  the idea of Mazzeo and Pacard by making use of the action of the group of rigid motions in Euclidean spaces.  We are able to construct CMC surfaces in euclidean three space with  (mostly) freely prescribed finite topology.