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Double bubble theorem

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

In the mathematical theory of minimal surfaces, the double bubble theorem 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 120° on a common circle. It was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989,[1] but was not proven until 2002. An analogous result on the optimal enclosure of two volumes generalizes to Euclidean spaces of any dimension.

Statement

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 120° ( radians).[2] In a standard double bubble, three patches of spheres meet at this angle along a shared circle. Two of these spherical surfaces form the outside boundary of the double bubble and a third one in the interior separates 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.[3] In the special case of two equal volumes, the middle surface is instead a flat disk, which can be interpreted as a patch of an infinite-radius sphere.

The double bubble theorem 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.[1]

Double bubbles in the Euclidean plane with three different combinations of areas. Rotating each of these in 3D, with its vertical symmetry line as the rotation axis, produces a three-dimensional double bubble as a surface of revolution.

In the Euclidean plane, analogously, the minimum perimeter of a system of curves that enclose two given areas is formed by three circular arcs, meeting at the same angle of 120°. For two equal areas, the middle arc degenerates to a straight line segment.[4] The three-dimensional standard double bubble can be seen as a surface of revolution of this two-dimensional double bubble.[5] In any higher dimension, the optimal enclosure for two volumes is again formed by three patches of hyperspheres, meeting at the same 120° angle.[1][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 theorem was assumed without proof by C. V. Boys in the 1912 edition of his book on soap bubbles.[7][8]

By 1989, the problem had become a "serious focus of research".[1] 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.[4][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.[5][9][12] The generalization to higher dimensions was published by Reichardt in 2008,[6] and in 2014, Lawlor published an alternative proof of the double bubble theorem generalizing both to higher dimensions and to weighted forms of surface energy.[1] Variations of the problem considering other measures of the size of the enclosing surface, such as its Gaussian measure, have also been studied.[13]

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.[14]

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 personal computer.[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][15]

Limiting shape of the curve-shortening flow for three regions, a degenerate planar double bubble with two infinite regions

John M. Sullivan has conjectured that, for any dimension , the minimum enclosure of up to volumes has the form of a stereographic projection of a simplex.[16] 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.[17] However, numerical experiments have shown that for six or more volumes in three dimensions, some of the boundaries between bubbles may be non-spherical.[16]

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, and its optimality (the honeycomb conjecture) was proven by T. C. Hales in 2001.[18] 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.[19]

Researchers have also studied the dynamics of physical processes by which pairs of bubbles coalesce into a double bubble.[20][21] This topic relates to a more general topic in differential geometry of the dynamic behavior of curves and surfaces under different processes that change them continuously. For instance, the curve-shortening flow is a process in which curves in the plane move at a speed proportionally to their curvature. It does not preserve areas of the regions enclosed by the curves, and the boundaries of bounded regions eventually vanish. For three regions, one of which is bounded and lies between the other two, which are separated by a line, the curve-shortening flow on their boundaries converges towards a limiting shape in the form of a degenerate double bubble: a vesica piscis along the line between the two unbounded regions.[22]

References

  1. ^ a b c d e Lawlor, Gary R. (2014), "Double bubbles for immiscible fluids in ", Journal of Geometric Analysis, 24 (1): 190–204, doi:10.1007/s12220-012-9333-1, MR 3145921
  2. ^ Taylor, Jean E. (1976), "The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces", Annals of Mathematics, 2nd Series, 103 (3): 489–539, doi:10.2307/1970949, JSTOR 1970949, MR 0428181
  3. ^ Isenberg, Cyril (1978), "Chapter 5. The Laplace–Young Equation", The Science of Soap Films and Soap Bubbles, Tieto Ltd, pp. 107–136; reprint, Dover Books, 1992, ISBN 0-486-26960-4
  4. ^ a b Alfaro, M.; Brock, J.; Foisy, J.; Hodges, N.; Zimba, J. (1993), "The standard double soap bubble in uniquely minimizes perimeter", Pacific Journal of Mathematics, 159 (1): 47–59, doi:10.2140/pjm.1993.159.47, MR 1211384
  5. ^ a b 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
  6. ^ a b Reichardt, Ben W. (2008), "Proof of the double bubble conjecture in ", Journal of Geometric Analysis, 18 (1): 172–191, arXiv:0705.1601, doi:10.1007/s12220-007-9002-y, MR 2365672
  7. ^ a b c d e Morgan, Frank (2004), "Proof of the double bubble conjecture", in Hardt, Robert (ed.), Six Themes on Variation, Student Mathematical Library, vol. 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. (1912), "Composite bubbles", Soap-Bubbles, Their Colours, And The Forces Which Mould Them, Society for Promoting Christian Knowledge, pp. 120–127
  9. ^ a b Devlin, Keith (22 March 2000), "Blowing out the bubble reputation: Four mathematicians have just cleaned up a long-standing conundrum set by soapy water", The Guardian
  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. ^ a b 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. ^ Milman, Emanuel; Neeman, Joe (2022), "The Gaussian double-bubble and multi-bubble conjectures", Annals of Mathematics, Second Series, 195 (1): 89–206, doi:10.4007/annals.2022.195.1.2, MR 4358414
  14. ^ Hutchings, Michael (1997), "The structure of area-minimizing double bubbles", Journal of Geometric Analysis, 7 (2): 285–304, doi:10.1007/BF02921724, MR 1646776
  15. ^ Morgan, Frank (2016), "Chapter 14. Proof of Double Bubble Conjecture", Geometric Measure Theory: A Beginner's Guide (5th ed.), Academic Press, pp. 143–158, ISBN 978-0-12-804527-5
  16. ^ a b 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., vol. 354, Dordrecht: Kluwer Acad. Publ., pp. 379–402, MR 1688327
  17. ^ 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
  18. ^ 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
  19. ^ 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
  20. ^ Besson, S.; Debrégeas, G. (October 2007), "Statics and dynamics of adhesion between two soap bubbles", The European Physical Journal E, 24 (2): 109–117, doi:10.1140/epje/i2007-10219-y
  21. ^ Ďurikovič, Roman (2001), "Animation of soap bubble dynamics, cluster formation and collision", Comput. Graph. Forum, 20 (3): 67–76, doi:10.1111/1467-8659.00499
  22. ^ Bellettini, Giovanni; Novaga, Matteo (2011), "Curvature evolution of nonconvex lens-shaped domains", Journal für die Reine und Angewandte Mathematik, 2011 (656): 17–46, arXiv:0906.0166, doi:10.1515/CRELLE.2011.041, MR 2818854, S2CID 14158286
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