We will discuss recent work in which we construct complete embedded constant mean curvature surfaces in \(\mathbb{R}^3\) with freely prescribed genus and any number of ends greater than or equal to three. It has been known since1989 that complete embedded constant mean curvatures surfaces have at two ends, and in the two ended case the only embedded surfaces are Delaunay ends. Thus, our theorem is then optimal in the sense that all possible topologies are realized. Prior to this, Kapouleas and Briener constructed surfaces with arbitrarily many ends greater than two, however for each number of ends, only one topological type is realized. The construction is derived from a gluing technique introduced by Mazzeo and Pacard in the 90’s which uses a Cauchy data matching procedure, and ours offers several refinements. In this talk, we will give a general overview of the history of the problem and the techniques involved.

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