## Proof of the double bubble conjecture.(English)Zbl 0970.53009

The authors announce a proof of the double bubble conjecture: “the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in $$\mathbb{R}^3$$.” After discussing the historic background and previous results they give an outline of their proof, that is available as a preprint from the third author’s homepage. For a related result see [J. Hass, R. Schlafly, Ann. Math. 151, 459-515 (2000; Zbl 0970.53008)] the preceding review.

### MSC:

 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 76B45 Capillarity (surface tension) for incompressible inviscid fluids 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 49Q10 Optimization of shapes other than minimal surfaces

### Keywords:

double bubble; soap bubble; isoperimetric problem; stability

Zbl 0970.53008
Full Text:

### References:

 [1] 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. · Zbl 0327.49043 [2] C. V. Boys, Soap-Bubbles, Dover, New York, 1959. [3] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Interscience Publishers, New York, 1953. · Zbl 0051.28802 [4] Joel Foisy, Soap bubble clusters in $${\mathbb R}^2$$ and $${\mathbb R}^3$$, undergraduate thesis, Williams College, 1991. [5] Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba, The standard double soap bubble in \?² uniquely minimizes perimeter, Pacific J. Math. 159 (1993), no. 1, 47 – 59. · Zbl 0738.49023 [6] Joel Hass, Michael Hutchings, and Roger Schlafly, The double bubble conjecture, Electron. Res. Announc. Amer. Math. Soc. 1 (1995), no. 3, 98 – 102. · Zbl 0864.53007 [7] Joel Hass and Roger Schlafly, Bubbles and double bubbles, American Scientist, Sept-Oct 1996, 462-467. · Zbl 0970.53008 [8] Joel Hass and Roger Schlafly, Double bubbles minimize, Annals of Mathematics 151 (2000), 459-515. · Zbl 0970.53008 [9] Michael Hutchings, The structure of area-minimizing double bubbles, J. Geom. Anal. 7 (1997), no. 2, 285 – 304. · Zbl 0935.53008 [10] 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. · Zbl 1009.53007 [11] Wilbur Richard Knorr, The ancient tradition of geometric problems, Birkhäuser Boston, Inc., Boston, MA, 1986. · Zbl 0588.01002 [12] Frank Morgan, The double bubble conjecture, FOCUS, Math. Assn. Amer., December, 1995. · Zbl 0990.49029 [13] Frank Morgan, Geometric measure theory, 2nd ed., Academic Press, Inc., San Diego, CA, 1995. A beginner’s guide. · Zbl 0819.49024 [14] 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. · Zbl 0956.53049 [15] J. Plateau, Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires, Paris, Gauthier-Villars, 1873. · JFM 06.0516.03 [16] 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). · Zbl 1056.53007 [17] 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. · Zbl 0760.53037 [18] Antonio Ros and Rabah Souam, On stability of capillary surfaces in a ball, Pacific J. Math. 178 (1997), no. 2, 345 – 361. · Zbl 0930.53007 [19] Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1995), no. 1, 19 – 33. · Zbl 0912.53009 [20] 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. · JFM 16.0232.04 [21] 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. , https://doi.org/10.2307/1970949 Jean E. Taylor, The structure of singularities in solutions to ellipsoidal variational problems with constraints in \?³, Ann. of Math. (2) 103 (1976), no. 3, 541 – 546. · Zbl 0335.49033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.