Proof of the double bubble conjecture
type: Item
instance of: scholarly article
title: Proof of the double bubble conjecture
zbMATH Open document ID: 1009.53007
DOI: 10.2307/3062123
author: Michael Hutchings
author: Frank Morgan
author: Manuel Ritoré
author: Antonio Ros
published in: Annals of Mathematics. Second Series
publication date: 2003-04-04T00:00:00Z
full work available at URL: http://hdl.handle.net/10481/32449
review text: The double bubble conjecture (now theorem) is 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 \(v_2\) are 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 conjecture was observed and published by Plateau in 1873 (see JFM 06.0516.03). One of the difficulties is that existence proofs rely on allowing the regions \(R_1\) and \(R_2\) to be disconnected. A priori even the exterior region complementary to \(R_1\) and \(R_2\) might be disconnected. In this paper, an estimate of Hutchings is used to prove that the larger region is connected. A stability argument is used to show that the smaller region has at most two components. A second crucial part of the argument involves the consideration of rotations about an axis orthogonal to the axis of symmetry of the double bubble. The proper choice of axis allows the construction of variations that respect both volume constraints. Stability implies that the variation satisfies a nice differential equation leading to sufficient information to conclude that the surface is made up of pieces of spheres.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1779740
zbMATH Keywords: double bubble
zbMATH Keywords: prescibed volume
zbMATH Keywords: least-area
reviewed by: Harold Parks
MaRDI profile type: Publication
OpenAlex ID: W2612390739
Wikidata QID: Q112572095
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description: scientific article; zbMATH DE number 1779740