## Triunduloids

### Embedded Constant Mean Curvature Surfaces with Three Ends and Genus Zero

**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.

- triund.pdf
- 24-page PDF version of the paper, with figures

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.