Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



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
Published electronically: July 17, 2000
MathSciNet review: 1777854
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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

References [Enhancements On Off] (What's this?)

  • [A] F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs AMS 4 (1976), no. 165. MR 54:8420
  • [B] C. V. Boys, Soap-Bubbles, Dover, New York, 1959.
  • [CH] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Interscience Publishers, New York, 1953. MR 16:426a
  • [F1] Joel Foisy, Soap bubble clusters in ${\mathbb R}^2$ and ${\mathbb R}^3$, undergraduate thesis, Williams College, 1991.
  • [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. MR 94b:53019
  • [HHS] Joel Hass, Michael Hutchings, and Roger Schlafly, The double bubble conjecture, Elec. Res. Ann. AMS 1 (1995), 98-102. MR 97b:53014
  • [HS1] Joel Hass and Roger Schlafly, Bubbles and double bubbles, American Scientist, Sept-Oct 1996, 462-467.
  • [HS2] Joel Hass and Roger Schlafly, Double bubbles minimize, Annals of Mathematics 151 (2000), 459-515.
  • [Hu] Michael Hutchings, The structure of area-minimizing double bubbles, J. Geom. Anal. 7 (1997), 285-304. MR 99j:53010
  • [HMRR] Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture, preprint (2000), available at$\sim$ritore/bubble/ bubble.htm.
  • [K] Wilbur Richard Knorr, The ancient tradition of geometric problems, Birkhäuser, Boston, 1986. MR 88e:01010
  • [M1] Frank Morgan, The double bubble conjecture, FOCUS, Math. Assn. Amer., December, 1995.
  • [M2] Frank Morgan, Geometric measure theory: a beginner's guide, third edition, Academic Press, 2000. MR 96c:49001
  • [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] J. Plateau, Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires, Paris, Gauthier-Villars, 1873.
  • [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] 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. MR 93a:53055
  • [RS] Antonio Ros and Rabah Souam,On stability of capillary surfaces, Pacific J. Math 178 (1997), 345-361. MR 98c:58029
  • [RV] Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1995), 19-33. MR 96h:53013
  • [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] Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. Math. 103 (1976), 489-539. MR 55:1208a

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 53A10, 53C42

Retrieve articles in all journals with MSC (2000): 53A10, 53C42

Additional Information

Michael Hutchings
Affiliation: Department of Mathematics, Stanford University, Stanford, CA 94305

Frank Morgan
Affiliation: Department of Mathematics, Williams College, Williamstown, MA 01267

Manuel Ritoré
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España

Antonio Ros
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España

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

American Mathematical Society