### Constant mean curvature surfaces with cylindrical ends

Published in **Mathematical Visualization**,
(H.-C. Hege, K. Polthier, eds.), Springer, 1998, pp 107-116.
**Plain-English Abstract**
Due to surface tension, a soap bubble (or any interface in a soap
froth) minimizes its area while keeping a fixed enclosed volume.
The solution to such a problem, by Laplace's Law, is an embedded
surface with constant mean curvature, equal to the pressure difference
across the surface. Mathematically is it most natural to study such
"CMC" surfaces which are complete (have no boundary), although except
for the round sphere of a single soap bubble, the complete surfaces all
extend off to infinity in various directions, and thus could not be
physically stable soap films.

In the special case when there is no pressure difference across
the surface, the film is merely minimizing its area, and the
resulting surface is called a minimal surface. In the theory
of minimal surfaces, there has been a very useful distinction
between those of finite and infinite absolute total curvature.

We consider instead the case where the pressure difference is
nonzero. Here, Schoen had asked whether the sphere and the cylinder
are the only complete (almost) embedded CMC surfaces with finite
absolute total curvature. We propose an infinite family of further
such surfaces. The existence of these examples is supported by the
results of computer experiments.