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
Full-text PDF Free Access
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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 ${\mathbb R}^3$.
- 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
[A]A F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs AMS 4 (1976), no. 165.
[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.
[F2]F2 Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba, The standard double soap bubble in ${\mathbb R}^2$ uniquely minimizes perimeter, Pacific J. Math. 159 (1993), 47–59.
[HHS]HHS Joel Hass, Michael Hutchings, and Roger Schlafly, The double bubble conjecture, Elec. Res. Ann. AMS 1 (1995), 98–102.
[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.
[Hu]Hu Michael Hutchings, The structure of area-minimizing double bubbles, J. Geom. Anal. 7 (1997), 285–304.
[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.
[K]K Wilbur Richard Knorr, The ancient tradition of geometric problems, Birkhäuser, Boston, 1986.
[M1]M1 Frank Morgan, The double bubble conjecture, FOCUS, Math. Assn. Amer., December, 1995.
[M2]M2 Frank Morgan, Geometric measure theory: a beginner’s guide, third edition, Academic Press, 2000.
[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).
[RR]RR Manuel Ritoré and Antonio Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comment. Math. Helv. 67 (1992), 293–305.
[RS]RS Antonio Ros and Rabah Souam,On stability of capillary surfaces, Pacific J. Math 178 (1997), 345–361.
[RV]RV Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1995), 19–33.
[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.
[T]T Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. Math. 103 (1976), 489–539.
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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