The double bubble problem on the flat two-torus

Corneli, Joseph; Holt, Paul; Lee, George; Leger, Nicholas; Schoenfeld, Eric; Steinhurst, Benjamin

doi:10.1090/S0002-9947-04-03551-2

The double bubble problem on the flat two-torus
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by Joseph Corneli, Paul Holt, George Lee, Nicholas Leger, Eric Schoenfeld and Benjamin Steinhurst

Trans. Amer. Math. Soc. 356 (2004), 3769-3820

DOI: https://doi.org/10.1090/S0002-9947-04-03551-2

Published electronically: March 12, 2004

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Abstract:

We characterize the perimeter-minimizing double bubbles on all flat two-tori and, as corollaries, on the flat infinite cylinder and the flat infinite strip with free boundary. Specifically, we show that there are five distinct types of minimizers on flat two-tori, depending on the areas to be enclosed.

References

Miguel Carrión Álvarez, Joseph Corneli, Geneveive Walsh, and Shabnam Beheshti, Double bubbles in the three-torus, Exp. Math., 12 (2003), 79-89.
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
L. Fejes Tóth, Regular figures, A Pergamon Press Book, The Macmillan Company, New York, 1964. MR 0165423
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
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. 1, 47–59. MR 1211384, DOI 10.2140/pjm.1993.159.47
T. C. Hales, The honeycomb conjecture, Discrete Comput. Geom. 25 (2001), no. 1, 1–22. MR 1797293, DOI 10.1007/s004540010071
Hugh Howards, Michael Hutchings, and Frank Morgan, The isoperimetric problem on surfaces, Amer. Math. Monthly 106 (1999), no. 5, 430–439. MR 1699261, DOI 10.2307/2589147
Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture, Ann. of Math. (2) 155 (2002), no. 2, 459–489. MR 1906593, DOI 10.2307/3062123
Joseph D. Masters, The double bubble on the flat torus, Unpublished notes on file with F. Morgan, Williams College, 1994.
Joseph D. Masters, The perimeter-minimizing enclosure of two areas in $S^2$, Real Anal. Exchange 22 (1996/97), no. 2, 645–654. MR 1460978, DOI 10.2307/44153944
Frank Morgan, Geometric measure theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner’s guide. MR 1775760
Frank Morgan, Soap bubbles in $\textbf {R}^2$ and in surfaces, Pacific J. Math. 165 (1994), no. 2, 347–361. MR 1300837, DOI 10.2140/pjm.1994.165.347
Frank Morgan and Wacharin Wichiramala, The standard double bubble is the unique stable double bubble in $\mathbf R^2$, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2745–2751. MR 1900881, DOI 10.1090/S0002-9939-02-06640-6
Ben W. Reichardt, Cory Heilmann, Yuan Y. Lai, and Anita Spielman, Proof of the double bubble conjecture in $\mathbf {R}^4$ and certain higher dimensional cases, Pac. J. Math., 208 (2003), 347-366.
Wacharin Wichiramala, The planar triple bubble problem, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2002.