The Journal of Geometric Analysis

, Volume 17, Issue 2, pp 189–212 | Cite as

Double bubbles inS 3 andH 3

  • Joseph Corneli
  • Neil Hoffman
  • Paul Holt
  • George Lee
  • Nicholas Leger
  • Stephen Moseley
  • Eric Schoenfeld


We prove the double bubble conjecture in the three-sphereS 3 and hyperbolic three-spaceH 3 in the cases where we can apply Hutchings theory:
  • • InS 3, when each enclosed volume and the complement occupy at least 10% of the volume ofS 3.

  • • inH 3, when the smaller volume is at least 85% that of the larger.

A balancing argument and asymptotic analysis reduce the problem inS 3 andH 3 to some computer checking. The computer analysis has been designed and fully implemented for both spaces.

Math Subject Classifications


Key Words and Phrases

Double bubbles 


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  1. [1]
    Álvarez, M. C., Corneli, J., Walsh, G., and Beheshti, S. Double bubbles in the three-torus,J. Experimental Math. 12, 79–89, (2000).Google Scholar
  2. [2]
    Corneli, J. Double bubbles in spaces of constant curvature, undergraduate thesis, New College of Florida, 2002.Google Scholar
  3. [3]
    Corneli, J., Corwin, I., Hurder, S., Sesum, V., Xu, Y., Adams, E., Davis, D., Lee, M., and Visocchi, R. Double bubbles in Gauss space and spheres,Houston J. Math., to appear.Google Scholar
  4. [4]
    Corneli, J., Hoffman, N., Holt, P., Lee, G., Leger, N., Moseley, S., and Schoenfeld, E. Double Bubbles inS 3 andH 3, to appear at Scholar
  5. [5]
    Corneli, J., Hoffman, N. and Moseley, S. Double bubbles inS 3 andH 3 andG m, Williams College NSF SMALL undergraduate research Geometry Group report, 2003.Google Scholar
  6. [6]
    Corneli, J., Holt, P., Leger, N., and Schoenfeld, E. Double bubbles inS 3 andH 3, Williams College NSF SMALL undergraduate research Geometry Group report, 2001.Google Scholar
  7. [7]
    Cotton, A. and Freeman, D. The double bubble problem in spherical space and hyperbolic space.Int. Math. J. 32, 461–499, (2002).MathSciNetGoogle Scholar
  8. [8]
    Heilman, C., Lai, Y., Reichardt, B., and Spielman, A. Component bounds for area-minimizing double bubbles, NSF “SMALL” undergraduate research Geometry Group report, Williams College, (Chapter 14), 1999.Google Scholar
  9. [9]
    Hoffman, N. Double Bubbles inS 3,H 3 and Gauss Space, undergraduate thesis, Williams College, 2004.Google Scholar
  10. [10]
    Hutchings, M. The structure of area-minimizing double bubbles.J. Geom. Anal. 7(2), 285–304, (1997).zbMATHMathSciNetGoogle Scholar
  11. [11]
    Hutchings, M., Morgan, F., Ritoré, M., and Ros, A. Proof of the double bubble conjecture,Ann. of Math. (2)155, 459–489, (2000).CrossRefGoogle Scholar
  12. [12]
    Morgan, F.Geometric Measure Theory: A Beginner’s Guide, 3rd ed., Academic Press, San Diego CA, 2000.zbMATHGoogle Scholar
  13. [13]
    Reichardt, B. W., Heilmann, C., Lai, Y. Y., and Spielmann, A. Proof of the double bubble conjecture inR 4 and certain higher-dimensional cases.Pacific J. Math. 208, 347–366, (2003).zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    Schmidt E. Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionenzahl,Math. Z. 49, 1–109, (1943).zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Wolfram, S.Mathematica Version 5, Wolfram Research, Champaign, 2003.Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2007

Authors and Affiliations

  • Joseph Corneli
    • 1
  • Neil Hoffman
    • 1
  • Paul Holt
    • 2
  • George Lee
    • 2
  • Nicholas Leger
    • 1
  • Stephen Moseley
    • 3
  • Eric Schoenfeld
    • 4
  1. 1.Department of MathematicsUniversity of TexasAustin
  2. 2.Department of Mathematics and StatisticsWilliams CollegeWilliamstown
  3. 3.Center for Applied MathematicsCornell UniversityIthaca
  4. 4.Stanford University, MathematicsStanford

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