Abstract: In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; they are classified using their asymptotic necksizes. We work in a class slightly more general than embedded surfaces, namely immersed surfaces which bound an immersed three-manifold, as introduced by Alexandrov.
This paper appeared (2003) in Crelle's Journal für die reine und angewandte Mathematik.
Our brief 1999 announcement of this result was published in Proc. Natl. Acad. Sci USA 97:26 (Dec. 2000), pp. 14067-14068. The published version is available here as a two-page PDF file.
Our first preprint on the subject (from 1998) has more phenonenological description of the family than was retained in our later papers, so it still might be of some interest. (20 page PDF)
In November 2004, I gave a talk at the Fukuoka workshop on Integrable Systems, Geometry and Visualization. My 9-page report for the proceedings shares the title of my talk: A Complete Family of CMC Surfaces.