Double Bubbles in the Three-Torus

The classical isoperimetric problem seeks the least-area way to enclose a single region of prescribed volume. About 200 BC, Zenodorus argued that a circle is the least-perimeter enclosure of prescribed area in the plane (see [Heath 60]). In 1884, Schwarz [Schwarz 1884] proved by symmetrization that a sphere minimizes surface area for a given volume in R. Isoperimetric problems arise naturally in many parts of modern mathematics; Ros provides a nice survey [Ros 01]. The double bubble problem is a two-volume generalization of the classical isoperimetric problem. In 2000, Hutchings, Morgan, Ritoré, and Ros proved that the standard double bubble was the unique least-area way to enclose and separate two volumes in R ([Hutchings et al. 00], [Hutchings et al. 02], [Morgan 00, Chapter 14]). In this paper, we present a conjecture, based on substantial numerical evidence, on the least-area way to enclose and separate two volumes in the three-torus.