Double bubble conjecture

In the mathematical theory of minimal surfaces, the double bubble conjecture states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a standard double bubble — three spherical surfaces meeting at angles of 2π/3 on a common circle. It is now a theorem, as a proof of it was published in 2002.[1][2]

A double bubble. Note that the surface separating the small lower bubble from the large bubble bulges into the large bubble.

The conjecture

According to Plateau's laws, the minimum area shape that encloses any volume or set of volumes must take a form commonly seen in soap bubbles in which surfaces of constant mean curvature meet in threes, forming dihedral angles of 2π/3.[3] In a standard double bubble, these surfaces are patches of spheres, and the curve where they meet is a circle. When the two enclosed volumes are different from each other, there are three spherical surfaces, two on the outside of the double bubble and one in the interior, separating the two volumes from each other; the radii of the spheres is inversely proportional to the pressure differences between the volumes they separate, according to the Young–Laplace equation.[4] When the two volumes are equal, the middle surface is instead a flat disk, which can be interpreted as a patch of an infinite-radius sphere.

The double bubble conjecture states that, for any two volumes, the standard double bubble is the minimum area shape that encloses them; no other set of surfaces encloses the same amount of space with less total area.

The same fact is also true for the minimum-length set of curves in the Euclidean plane that encloses a given pair of areas,[5] and it can be generalized to any higher dimension.[6]

History

The isoperimetric inequality for three dimensions states that the shape enclosing the minimum single volume for its surface area is the sphere; it was formulated by Archimedes but not proven rigorously until the 19th century, by Hermann Schwarz. In the 19th century, Joseph Plateau studied the double bubble, and the truth of the double bubble conjecture was assumed without proof by C. V. Boys in his 1896 book on soap bubbles.[7][8]

In 1991, Joel Foisy, an undergraduate student at Williams College, was the leader of a team of undergraduates that proved the two-dimensional analogue of the double bubble conjecture.[5][7] In his undergraduate thesis, Foisy was the first to provide a precise statement of the three-dimensional double bubble conjecture, but he was unable to prove it.[9]

A proof for the restricted case of the double bubble conjecture, for two equal volumes, was announced by Joel Hass and Roger Schlafly in 1995, and published in 2000.[10][11] The proof of the full conjecture by Hutchings, Morgan, Ritoré, and Ros was announced in 2000 and published in 2002.[1][9][12]

Proof

A lemma of Brian White shows that the minimum area double bubble must be a surface of revolution. For, if not, it would be possible to find two orthogonal planes that bisect both volumes, replace surfaces in two of the four quadrants by the reflections of the surfaces in the other quadrants, and then smooth the singularities at the reflection planes, reducing the total area.[7] Based on this lemma, Michael Hutchings was able to restrict the possible shapes of non-standard optimal double bubbles, to consist of layers of toroidal tubes.[13]

Additionally, Hutchings showed that the number of toroids in a non-standard but minimizing double bubble could be bounded by a function of the two volumes. In particular, for two equal volumes, the only possible nonstandard double bubble consists of a single central bubble with a single toroid around its equator. Based on this simplification of the problem, Joel Hass and Roger Schlafly were able to reduce the proof of this case of the double bubble conjecture to a large computerized case analysis, taking 20 minutes on a 1995 PC.[7][11]

The eventual proof of the full double bubble conjecture also uses Hutchings' method to reduce the problem to a finite case analysis, but it avoids the use of computer calculations, and instead works by showing that all possible nonstandard double bubbles are unstable: they can be perturbed by arbitrarily small amounts to produce another solution with lower cost. The perturbations needed to prove this result are a carefully chosen set of rotations.[7]

John M. Sullivan has conjectured that, for any dimension d, the minimum enclosure of up to d + 1 volumes has the form of a stereographic projection of a simplex.[14] In particular, in this case, all boundaries between bubbles would be patches of spheres. The special case of this conjecture for three bubbles in two dimensions has been proven; in this case, the three bubbles are formed by six circular arcs and straight line segments, meeting in the same combinatorial pattern as the edges of a tetrahedron.[15] However, numerical experiments have shown that for six or more volumes in three dimensions, some of the boundaries between bubbles may be non-spherical.[14]

For an infinite number of equal areas in the plane, the minimum-length set of curves separating these areas is the hexagonal tiling, familiar from its use by bees to form honeycombs.[16] For the same problem in three dimensions, the optimal solution is not known; Lord Kelvin conjectured that it was given by a structure combinatorially equivalent to the bitruncated cubic honeycomb, but this conjecture was disproved by the discovery of the Weaire–Phelan structure, a partition of space into equal volume cells of two different shapes using a smaller average amount of surface area per cell.[17]

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References
  1. Hutchings, Michael; Morgan, Frank; Ritoré, Manuel; Ros, Antonio (2002), "Proof of the double bubble conjecture", Annals of Mathematics, 2nd Ser., 155 (2): 459–489, arXiv:math/0406017, doi:10.2307/3062123, JSTOR 3062123, MR 1906593.
  2. Morgan, Frank (2009), "Chapter 14. Proof of Double Bubble Conjecture", Geometric Measure Theory: A Beginner's Guide (4th ed.), Academic Press.
  3. Taylor, Jean E. (1976), "The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces", Annals of Mathematics, 2nd Ser., 103 (3): 489–539, doi:10.2307/1970949, JSTOR 1970949, MR 0428181.
  4. Isenberg, Cyril (1978), "Chapter 5. The Laplace–Young Equation", The Science of Soap Films and Soap Bubbles, Dover.
  5. Alfaro, M.; Brock, J.; Foisy, J.; Hodges, N.; Zimba, J. (1993), "The standard double soap bubble in R2 uniquely minimizes perimeter", Pacific Journal of Mathematics, 159 (1): 47–59, doi:10.2140/pjm.1993.159.47, MR 1211384.
  6. Reichardt, Ben W. (2008), "Proof of the double bubble conjecture in Rn", Journal of Geometric Analysis, 18 (1): 172–191, arXiv:0705.1601, doi:10.1007/s12220-007-9002-y, MR 2365672.
  7. Morgan, Frank (2004), "Proof of the double bubble conjecture", in Hardt, Robert (ed.), Six Themes on Variation, Student Mathematical Library, 26, American Mathematical Society, pp. 59–77, doi:10.1090/stml/026/04, hdl:10481/32449, MR 2108996. Revised version of an article initially appearing in the American Mathematical Monthly (2001), doi:10.2307/2695380, MR1834699.
  8. Boys, C. V. (1896), Soap-Bubbles And The Forces Which Mould Them, Society for Promoting Christian Knowledge.
  9. "Blowing out the bubble reputation: Four mathematicians have just cleaned up a long-standing conundrum set by soapy water, writes Keith Devlin", The Guardian, 22 March 2000.
  10. Peterson, Ivars (August 12, 1995), "Toil and trouble over double bubbles" (PDF), Science News, 148 (7): 101–102, doi:10.2307/3979333, JSTOR 3979333.
  11. Hass, Joel; Schlafly, Roger (2000), "Double bubbles minimize", Annals of Mathematics, 2nd Ser., 151 (2): 459–515, arXiv:math/0003157, Bibcode:2000math......3157H, doi:10.2307/121042, JSTOR 121042, MR 1765704. Previously announced in Electronic Research Announcements of the American Mathematical Society, 1995, doi:10.1090/S1079-6762-95-03001-0.
  12. Cipra, Barry A. (March 17, 2000), "Mathematics: Why Double Bubbles Form the Way They Do", Science, 287 (5460): 1910–1912, doi:10.1126/science.287.5460.1910a
  13. Hutchings, Michael (1997), "The structure of area-minimizing double bubbles", Journal of Geometric Analysis, 7 (2): 285–304, doi:10.1007/BF02921724, MR 1646776.
  14. Sullivan, John M. (1999), "The geometry of bubbles and foams", in Sadoc, Jean-François; Rivier, Nicolas (eds.), Foams and Emulsions: Proc. NATO Advanced Study Inst. on Foams and Emulsions, Emulsions and Cellular Materials, Cargèse, Corsica, 12–24 May, 1997, NATO Adv. Sci. Inst. Ser. E Appl. Sci., 354, Dordrecht: Kluwer Acad. Publ., pp. 379–402, MR 1688327.
  15. Wichiramala, Wacharin (2004), "Proof of the planar triple bubble conjecture", Journal für die Reine und Angewandte Mathematik, 2004 (567): 1–49, doi:10.1515/crll.2004.011, MR 2038304.
  16. Hales, Thomas C. (2001), "The honeycomb conjecture", Discrete and Computational Geometry, 25 (1): 1–22, arXiv:math.MG/9906042, doi:10.1007/s004540010071, MR 1797293.
  17. Weaire, Denis; Phelan, Robert (1994), "A counter-example to Kelvin's conjecture on minimal surfaces", Philosophical Magazine Letters, 69 (2): 107–110, Bibcode:1994PMagL..69..107W, doi:10.1080/09500839408241577.
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