Constant mean curvature surfaces with three ends

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

Published in Proc. Natl. Acad. Sci. USA 97:26, 2000 Dec 19, pp 14067-14068. 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 (the "ends" of the surface), and thus could not be physically stable soap films.

There are no embedded CMC surfaces with a single end, and the surfaces with two ends are all unduloids. These are the surfaces of revolution described by Delaunay in the last century, and are classified by their necksize.

Thus the first interesting case is that of CMC surfaces with three ends. This note announces our complete classification of such surfaces. Each end is asymptotic to an unduloid, so we call the surfaces triunduloids. We show that the moduli space of triunduloids is equivalent to the space of triples of points on the sphere, where the distances between the points are the necksizes of the three ends.