The standard double bubble is the unique stable double bubble in $\mathbf {R}^2$
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- by Frank Morgan and Wacharin Wichiramala PDF
- Proc. Amer. Math. Soc. 130 (2002), 2745-2751
Abstract:
We prove that the only equilibrium double bubble in $\mathbf {R}^2$ 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.References
- F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no.Β 165, viii+199. MR 420406, DOI 10.1090/memo/0165
- Andrew Cotton and David Freeman, The double bubble problem in spherical and hyperbolic space, preprint (2000).
- Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba, The standard double soap bubble in $\textbf {R}^2$ uniquely minimizes perimeter, Pacific J. Math. 159 (1993), no.Β 1, 47β59. MR 1211384
- Michael Hutchings, Frank Morgan, Manuel Ritore, and Antonio Ros, Proof of the double bubble conjecture, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 45β49. MR 1777854, DOI 10.1090/S1079-6762-00-00079-2
- Frank Morgan, Geometric measure theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginnerβs guide. MR 1775760
- Frank Morgan, $(\textbf {M},\epsilon ,\delta )$-minimal curve regularity, Proc. Amer. Math. Soc. 120 (1994), no.Β 3, 677β686. MR 1169884, DOI 10.1090/S0002-9939-1994-1169884-3
- β, 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 23β24, 2001, to appear.
- Frank Morgan, Soap bubbles in $\textbf {R}^2$ and in surfaces, Pacific J. Math. 165 (1994), no.Β 2, 347β361. MR 1300837
- Ben W. Reichardt, Cory Heilmann, Yuan Y. Lai, Anita Spielman, Proof of the double bubble conjecture in $\mathbf {R}^4$ and certain higher dimensional cases, Pacific J. Math., to appear.
- John M. Sullivan and Frank Morgan, Open problems in soap bubble geometry, Internat. J. Math. 7 (1996), no.Β 6, 833β842. MR 1417788, DOI 10.1142/S0129167X9600044X
- Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no.Β 3, 489β539. MR 428181, DOI 10.2307/1970949
- Wacharin Wichiramala, The planar triple bubble problem, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2002.
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
- Received by editor(s): April 18, 2001
- Published electronically: April 17, 2002
- Communicated by: Bennett Chow
- © Copyright 2002 by the authors
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2745-2751
- MSC (2000): Primary 53A10, 49Q20, 53Cxx
- DOI: https://doi.org/10.1090/S0002-9939-02-06640-6
- MathSciNet review: 1900881