Coplanar constant mean curvature surfaces

TL;DR

This paper extends the classification of genus-zero coplanar constant mean curvature surfaces in three-space to cases with any number of ends, assuming the ends' axes are coplanar, building on prior work for three ends.

Contribution

It generalizes previous classifications to an arbitrary number of ends under coplanarity assumptions, providing a complete family of such surfaces.

Findings

Constructed new families of surfaces with multiple ends

Extended classification results to more complex topologies

Provided explicit descriptions of the entire family of surfaces

Abstract

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.