The double bubble problem in spherical and hyperbolic space
type: Item
instance of: scholarly article
title: The double bubble problem in spherical and hyperbolic space
zbMATH Open document ID: 1036.49039
DOI: 10.1155/S0161171202207188
author: Yanyan Li
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2003-03-17T00:00:00Z
full work available at URL: https://eudml.org/doc/50739
review text: 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.
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 1883649
zbMATH Keywords: double bubble
zbMATH Keywords: hyperbolic space
zbMATH Keywords: spherical space
reviewed by: Harold Parks
MaRDI profile type: Publication
OpenAlex ID: W2125039390
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description: scientific article; zbMATH DE number 1883649