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The standard double bubble is the unique stable double bubble in 
Authors:
Frank Morgan and Wacharin Wichiramala
Journal:
Proc. Amer. Math. Soc. 130 (2002), 2745-2751
MSC (2000):
Primary 53A10, 49Q20, 53Cxx
Published electronically:
April 17, 2002
MathSciNet review:
1900881
Full-text PDF Free Access
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Additional Information
Abstract: We prove that the only equilibrium double bubble in  which is stable for fixed areas is the standard double bubble. This uniqueness result also holds for small stable double bubbles in surfaces, where it is new even for perimeter-minimizing double bubbles.
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Andrew Cotton and David Freeman, The double bubble problem in spherical and hyperbolic space, preprint (2000).
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Soc. 6 (2000),
45–49. MR
1777854 (2001m:53011), http://dx.doi.org/10.1090/S1079-6762-00-00079-2
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Ben W. Reichardt, Cory Heilmann, Yuan Y. Lai, Anita Spielman, Proof of the double bubble conjecture in
 and certain higher dimensional cases, Pacific J. Math., to appear.
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John
M. Sullivan and Frank
Morgan, Open problems in soap bubble geometry, Internat. J.
Math. 7 (1996), no. 6, 833–842. MR 1417788
(98a:53014), http://dx.doi.org/10.1142/S0129167X9600044X
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soap-film-like minimal surfaces, Ann. of Math. (2)
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Wacharin Wichiramala, The planar triple bubble problem, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2002.
- [A]
- F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variation problems with constraints, Mem. AMS No. 165 (1976). MR 54:8420
- [CF]
- Andrew Cotton and David Freeman, The double bubble problem in spherical and hyperbolic space, preprint (2000).
- [F]
- Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba, The standard double soap bubble in
 uniquely minimizes perimeter, Pacific J. Math. 159 (1993), 47-59. MR 94b:53019
- [HMRR]
- Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the Double Bubble Conjecture, Ann. Math. 155 (March, 2002), 459-489. Research announcement Electron. Res. Announce. Amer. Math. Soc. 6 (2000), 45-49. MR 2001m:53011
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- Frank Morgan, Geometric Measure Theory: a Beginner's Guide, third ed., Academic Press, 2000. MR 2001j:49001
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- -,
 -minimal curve regularity, Proc. AMS 120 (1994), 677-686. MR 94e:49018
- [M3]
- -, Small perimeter-minimizing double bubbles in compact surfaces are standard, Electronic Proceedings of the 78th annual meeting of the Louisiana/Mississippi Section of the MAA, Univ. of Miss., March 23-24, 2001, to appear.
- [M4]
- -, Soap bubbles in
 and in surfaces, Pac. J. Math. 165 (1994), 347-361. MR 96a:58064
- [RHLS]
- Ben W. Reichardt, Cory Heilmann, Yuan Y. Lai, Anita Spielman, Proof of the double bubble conjecture in and certain higher dimensional cases, Pacific J. Math., to appear.
- [SM]
- John M. Sullivan and Frank Morgan, ed., Open problems in soap bubble geometry, International J. Math. 7 (1996), 833-842. MR 98a:53014
- [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
- [W]
- Wacharin Wichiramala, The planar triple bubble problem, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2002.
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Additional Information
Frank Morgan
Affiliation:
Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267
Email:
Frank.Morgan@williams.edu
Wacharin Wichiramala
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email:
wichiram@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06640-6
PII:
S 0002-9939(02)06640-6
Keywords:
Stable double bubble,
standard double bubble,
soap bubble
Received by editor(s):
April 18, 2001
Published electronically:
April 17, 2002
Communicated by:
Bennett Chow
Article copyright:
© Copyright 2002
by the authors
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