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In 1841, Delaunay constructed the embedded surfaces of revolution withconstant mean curvature (CMC); these unduloids have genus zero and are nowknown 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 endsand genus zero; they are classified using their asymptotic necksizes. We workin a class slightly more general than embedded surfaces, namely immersedsurfaces which bound an immersed three-manifold, as introduced by Alexandrov.

Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero