Coplanar constant mean curvature surfaces
  We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in arXiv:math.DG/0102183. Here we extend the arguments to the case of an arbitrary number of ends, under the assumption that the asymptotic axes of the ends lie in a common plane: we construct and classify the entire family of these genus-zero coplanar constant mean curvature surfaces. 
Karsten Grosse-Brauckmann (TU Darmstadt), Robert B. Kusner (UMass   Amherst), John M. Sullivan (TU Berlin)