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Double bubbles in the three-torus. (English) Zbl 1061.53005
The authors treat the problem of enclosing and separating two given volumes in flat \(3\)-tori with the least possible perimeter. Minimizers for this problem are called double bubbles. As they remark in the introduction, the problem of enclosing one given volume with the least possible perimeter in a three-torus is not yet completely understood.
The authors first state conjecture 2.1, based on numerical evidence provided by Brakke’s surface evolver, on the possible minimizers in the cubic \(3\)-torus. The ten candidates are described in the second section and listed in figure 1 of the paper. In addition, they prove in theorem 4.1 that the standard double bubble is the least-perimeter way of enclosing and separating two small given volumes in a \(3\) or \(4\)-dimensional flat Riemannian manifold which has compact quotient by its isometry group. In their proof the classification of double bubbles in \({\mathbb R}^3\) [M. Hutchings, F. Morgan, M. Ritoré, and A. Ros, Ann. Math. (2) 155, 459–489 (2002; Zbl 1009.53007)] and \({\mathbb R}^4\) [B. W. Reichardt, C. Heilmann, Y. L. Lai and A. Spielman, Pac. J. Math. 208, 347–366 (2003; Zbl 1056.53007)] is used.
In the third and fifth sections of the paper, some related conjectures are stated.

MSC:
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q20 Variational problems in a geometric measure-theoretic setting
49Q05 Minimal surfaces and optimization
Software:
Surface Evolver
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References:
[1] Adams Colin C., ”When Soap Bubbles Collide.” (2003) · Zbl 1183.52015
[2] DOI: 10.2307/1970868 · Zbl 0252.49028
[3] DOI: 10.1016/S0246-0203(00)00131-X · Zbl 0964.60018
[4] Brakke Kenneth A., Surface Evolver, Version 2.14 (1999)
[5] Corneli Joseph, ”The Double Bubble Problem on the Flat Two-Torus.” (2003)
[6] Federer Herbert, Geometric Measure Theory. (1969)
[7] Heath Thomas, A History of Greek Mathematics (1960)
[8] Hurewicz Witold, Dimension Theory. (1941)
[9] DOI: 10.1007/BF02921724 · Zbl 0935.53008
[10] DOI: 10.1090/S1079-6762-00-00079-2 · Zbl 0970.53009
[11] DOI: 10.2307/3062123 · Zbl 1009.53007
[12] Kusner Robert B., Forma 11 pp 233– (1996)
[13] Morgan Frank, Geometric Measure Theory: A Beginner’s Guide, (2000) · Zbl 0974.49025
[14] Morgan, Frank. ”Small Perimeter-Minimizing Double Bubbles in Compact Surfaces are Standard.”. Electronic Proceedings of the 78th Annual meeting of the Lousiana/Mississippi Section of the MAA. March 23–24. Univ. of Miss. [Morgan 01], Available from World Wide Webhttp://www.mc.edu/campus/users/travis/maa/proceedings/spring2001/2001
[15] DOI: 10.1512/iumj.2000.49.1929 · Zbl 1021.53020
[16] Morgan Frank, ”Geometric Measure Theory and the Proof of the Double Bubble Conjecture.” (2001) · Zbl 0990.49029
[17] DOI: 10.1090/S0002-9939-02-06640-6 · Zbl 1003.53010
[18] DOI: 10.2140/pjm.2003.208.347 · Zbl 1056.53007
[19] DOI: 10.1007/PL00004351 · Zbl 0889.53045
[20] DOI: 10.1090/S0002-9947-96-01496-1 · Zbl 0867.53007
[21] Ros Antonio, ”The Isoperimetric Problem.” (2001) · Zbl 1005.53054
[22] Schwarz Hermann A., Nachrichten Königlichen Gesellschaft Wissenschaften Göttingen pp 1– (1884)
[23] Simon, Leon. ”Lectures on Geometric Measure Theory.”. Proc. Centre Math. Anal. Vol. 3, Canberra, Australia: Centre for Mathematical Analysis, Australian National University. [Simon 84], Australian Nat. U · Zbl 0546.49019
[24] DOI: 10.2307/1970949 · Zbl 0335.49032
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