Geodesics and soap bubbles in surfaces

Abstract

, 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 R 3 presumably provide the least-area way to enclose and separate the given volumes of air. Drawings by J. Bredt [M5] 347 348 FRANK MORGAN

Isoperimetric curves on hyperbolic surfaces

“…It it goes out a flared end, it can be translated back inside a fixed compact region. Hence the results of [HM,§3], stated for compact surfaces, apply and provide the asserted existence and regularity.…”

The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

“…The existence and regularity results of [HM,3.1,3.3] extend immediately to surfaces with convex boundary and provide a minimizer, possibly with bumping as asserted, C 1 away from ∂S, constant curvature κ 0 away from bumping. (As long as the boundary is convex, it does not interfere with the proof, a convexification argument.)…”

“…Lemma 2.2. Existence without an a priori bound on the number of boundary components follows as in [HM,Thm. 3.4].…”