The double bubble problem in spherical and hyperbolic space (Q1864248)

The double bubble theorem states that, in \({\mathbb R}^3\), the unique perimeter-minimizing double bubble enclosing and separating regions \(R_1\) and \(R_2\) of prescribed volumes \(v_1\) and \(v_2\) is a standard double bubble consisting of three spherical caps meeting along a common circle at 120-degree angles (for equal volumes, the middle cap is a flat disc). The physical fact expressed by the double bubble theorem was observed and published by Plateau in 1873. The paper under review extends the double bubble theorem to the equal volume case in the hyperbolic space \({\mathbb H}^3\) and extends the theorem to the equal volume case in the spherical space \(S^3\) provided that, additionally, the two volumes total to at most 90 percent of the volume of \(S^3\). It is also noted that the double bubble theorem holds in \({\mathbb H}^2\). In overall outline the proof is similar to that of \textit{M. Hutchings}, \textit{F. Morgan}, \textit{M. Ritoré} and \textit{A. Ros} [Ann. Math. (2) 155, 459--489 (2002; Zbl 1009.53007)], but certain equations and estimates particular to hyperbolic and spherical spaces are used.