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Double bubbles inS 3 andH 3

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Abstract

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.

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References

  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. Corneli, J. Double bubbles in spaces of constant curvature, undergraduate thesis, New College of Florida, 2002.

  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.

  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 arXiv.org.

  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.

  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.

  7. Cotton, A. and Freeman, D. The double bubble problem in spherical space and hyperbolic space.Int. Math. J. 32, 461–499, (2002).

    MathSciNet  Google Scholar 

  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.

  9. Hoffman, N. Double Bubbles inS 3,H 3 and Gauss Space, undergraduate thesis, Williams College, 2004.

  10. Hutchings, M. The structure of area-minimizing double bubbles.J. Geom. Anal. 7(2), 285–304, (1997).

    MATH  MathSciNet  Google Scholar 

  11. Hutchings, M., Morgan, F., Ritoré, M., and Ros, A. Proof of the double bubble conjecture,Ann. of Math. (2)155, 459–489, (2000).

    Article  Google Scholar 

  12. Morgan, F.Geometric Measure Theory: A Beginner’s Guide, 3rd ed., Academic Press, San Diego CA, 2000.

    MATH  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  14. Schmidt E. Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionenzahl,Math. Z. 49, 1–109, (1943).

    Article  MATH  MathSciNet  Google Scholar 

  15. Wolfram, S.Mathematica Version 5, Wolfram Research, Champaign, 2003.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas, 78712, Austin, TX

    Joseph Corneli, Neil Hoffman & Nicholas Leger

  2. Department of Mathematics and Statistics, Williams College, 01267, Williamstown, MA

    Paul Holt & George Lee

  3. Center for Applied Mathematics, Cornell University, 657 Frank H.T. Rhodes Hall, 14853, Ithaca, NY

    Stephen Moseley

  4. Stanford University, Mathematics, Bldg. 380, 450 Serra Mall, 94305-2125, Stanford, CA

    Eric Schoenfeld

Authors
  1. Joseph Corneli
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  2. Neil Hoffman
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  3. Paul Holt
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  4. George Lee
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  5. Nicholas Leger
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  6. Stephen Moseley
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  7. Eric Schoenfeld
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Correspondence to Joseph Corneli.

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Corneli, J., Hoffman, N., Holt, P. et al. Double bubbles inS 3 andH 3 . J Geom Anal 17, 189–212 (2007). https://doi.org/10.1007/BF02930720

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  • DOI: https://doi.org/10.1007/BF02930720

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