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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.

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
Surface Evolver
Full Text: DOI Euclid EuDML
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