Constant mean curvature surfaces with cylindrical ends

by Karsten Große-Brauckmann, Robert B. Kusner & John M. Sullivan

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.