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Proof of the double bubble conjecture


Authors: Michael Hutchings, Frank Morgan, Manuel Ritoré and Antonio Ros
Journal: Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 45-49
MSC (2000): Primary 53A10; Secondary 53C42
DOI: https://doi.org/10.1090/S1079-6762-00-00079-2
Published electronically: July 17, 2000
MathSciNet review: 1777854
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in ${\mathbb R}^3$.


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  • F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, viii+199. MR 420406, DOI https://doi.org/10.1090/memo/0165
  • [B]B C. V. Boys, Soap-Bubbles, Dover, New York, 1959. [CH]CH R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Interscience Publishers, New York, 1953. [F1]F1 Joel Foisy, Soap bubble clusters in ${\mathbb R}^2$ and ${\mathbb R}^3$, undergraduate thesis, Williams College, 1991.
  • Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba, The standard double soap bubble in ${\bf R}^2$ uniquely minimizes perimeter, Pacific J. Math. 159 (1993), no. 1, 47–59. MR 1211384
  • Joel Hass, Michael Hutchings, and Roger Schlafly, The double bubble conjecture, Electron. Res. Announc. Amer. Math. Soc. 1 (1995), no. 3, 98–102. MR 1369639, DOI https://doi.org/10.1090/S1079-6762-95-03001-0
  • [HS1]HS1 Joel Hass and Roger Schlafly, Bubbles and double bubbles, American Scientist, Sept–Oct 1996, 462–467. [HS2]HS2 Joel Hass and Roger Schlafly, Double bubbles minimize, Annals of Mathematics 151 (2000), 459–515.
  • Michael Hutchings, The structure of area-minimizing double bubbles, J. Geom. Anal. 7 (1997), no. 2, 285–304. MR 1646776, DOI https://doi.org/10.1007/BF02921724
  • [HMRR]HMRR Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture, preprint (2000), available at http://www.ugr.es/$\sim$ritore/bubble/ bubble.htm.
  • Wilbur Richard Knorr, The ancient tradition of geometric problems, Birkhäuser Boston, Inc., Boston, MA, 1986. MR 884893
  • [M1]M1 Frank Morgan, The double bubble conjecture, FOCUS, Math. Assn. Amer., December, 1995.
  • Frank Morgan, Geometric measure theory, 2nd ed., Academic Press, Inc., San Diego, CA, 1995. A beginner’s guide. MR 1326605
  • [PR]PR Renato H. L. Pedrosa and Manuel Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J., 48 (1999), 1357–1394. [P]P J. Plateau, Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires, Paris, Gauthier-Villars, 1873. [RHLS]RHLS Ben W. Reichardt, Cory Heilmann, Yuan Y. Lai, and Anita Spielman, Proof of the double bubble conjecture in ${\mathbf R}^4$ and certain higher dimensions, preprint (2000).
  • Manuel Ritoré and Antonio Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comment. Math. Helv. 67 (1992), no. 2, 293–305. MR 1161286, DOI https://doi.org/10.1007/BF02566501
  • Antonio Ros and Rabah Souam, On stability of capillary surfaces in a ball, Pacific J. Math. 178 (1997), no. 2, 345–361. MR 1447419, DOI https://doi.org/10.2140/pjm.1997.178.345
  • Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1995), no. 1, 19–33. MR 1338315, DOI https://doi.org/10.1007/BF01263611
  • [S]S H. A. Schwarz, Beweis des Satzes, dass die Kugel kleinere Oberfläche besitz, als jeder andere Körper gleichen Volumens, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1884), 1–13.
  • Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489–539. MR 428181, DOI https://doi.org/10.2307/1970949

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Additional Information

Michael Hutchings
Affiliation: Department of Mathematics, Stanford University, Stanford, CA 94305
Email: hutching@math.stanford.edu

Frank Morgan
Affiliation: Department of Mathematics, Williams College, Williamstown, MA 01267
Email: Frank.Morgan@williams.edu

Manuel Ritoré
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España
Email: ritore@ugr.es

Antonio Ros
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España
Email: aros@ugr.es

Keywords: Double bubble, soap bubbles, isoperimetric problems, stability
Received by editor(s): March 3, 2000
Published electronically: July 17, 2000
Communicated by: Richard Schoen
Article copyright: © Copyright 2000 Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros