Soap bubbles in \(\mathbb{R}^ 2\) and in surfaces (Q1343693)

The general soap bubble problem looks for a least-area way to enclose and separate a number of given volumes in \(\mathbb{R}^ n\) or in a smooth compact Riemannian manifold. Existence and regularity were treated by F. Almgren \((n \geq 3)\) and J. Taylor \((n = 3)\). The present paper proves the corresponding result in dimension 2: the solutions consist of constant- curvature arcs meeting in threes at 120 degrees. If the combinatorial type is described, too, then the arcs may bump up against each other.