Soap bubbles in ℝ² and in surfaces

Frank Morgan

Vol. 165 (1994), No. 2, 347–361

DOI: 10.2140/pjm.1994.165.347

Abstract

We prove existence and regularity for “soap bubbles” in ℝ² and in surfaces, i.e., the least-perimeter way to enclose and separate regions of prescribed area. They consist of constant-curvature arcs meeting in threes at 120 degrees. If one prescribes the combinatorial type too, then the arcs may bump up against each other.

Figure 1.1. Single, double, and triple bubbles in ℝ³ presumably provide the least-area way to enclose and separate the given volumes of air. Drawings by J. Bredt [M5]

Mathematical Subject Classification 2000

Primary: 58E12

Secondary: 49Q05, 53A10

Milestones

Received: 5 May 1992

Accepted: 14 June 1993

Published: 1 October 1994