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Isoperimetric surfaces with boundary


Authors: Rebecca Dorff, Drew Johnson, Gary R. Lawlor and Donald Sampson
Journal: Proc. Amer. Math. Soc. 139 (2011), 4467-4473
MSC (2010): Primary 53C38; Secondary 49Q10
DOI: https://doi.org/10.1090/S0002-9939-2011-10872-4
Published electronically: April 26, 2011
MathSciNet review: 2823092
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that many common combinations of soap films and soap bubbles that result from dipping polyhedral wire frames in a soap solution are minimizing with respect to their boundary and bubble volume. This can be thought of as a combination of the Plateau problem of least area for surfaces spanning a given boundary and the isoperimetric problem of least area for surfaces enclosing a given volume. Proof is given in arbitrary dimension using a combination of the mapping of Gromov, after Knothe, and the paired calibrations of Lawlor and Morgan.


References [Enhancements On Off] (What's this?)

  • 1. Yann Brenier.
    Polar factorization and monotone rearrangement of vector-valued functions.
    Comm. on Pure and Applied Math., 44:375-417, 1991. MR 1100809 (92d:46088)
  • 2. John Brothers and Frank Morgan.
    The isoperimetric theorem for general integrands.
    Michigan Math. J., 41(3):419-431, 1994. MR 1297699 (95g:49080)
  • 3. Jaigyoung Choe.
    Every stationary polyhedral set in $ R^n$ is area minimizing under diffeomorphisms.
    Pacific J. Math., 175(2):439-446, 1996. MR 1432839 (98e:49094)
  • 4. Rebecca Dorff, Gary Lawlor, Donald Sampson, and Brandon Wilson.
    Proof of the planar double bubble conjecture using metacalibration methods.
    Involve, 2(5):611-628, 2009. MR 2601581
  • 5. Michael Gromov.
    Isoperimetric inequalities in Riemannian manifolds, Appendix I to Asymptotic Theory of Finite Dimensional Normed Spaces by Vitali D. Milman and Gideon Schechtman, Lecture Notes in Mathematics, No. 1200. Springer-Verlag, New York, 1986. MR 0856576 (87m:46038)
  • 6. Reese Harvey and H. Blaine Lawson Jr.
    Calibrated geometries.
    Acta Mathematica, 148(1) (1982), 47-157. MR 0666108 (85i:53058)
  • 7. Herbert Knothe.
    Contributions to the theory of convex bodies.
    Michigan Math. J., (4):39-52, 1957. MR 0083759 (18:757b)
  • 8. Gary R. Lawlor and Frank Morgan.
    Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms.
    Pacific J. Math., 166(1):55-83, 1994. MR 1306034 (95i:58051)
  • 9. Frank Morgan.
    Geometric Measure Theory: A Beginner's Guide.
    Academic Press, 1988. MR 933756 (89f:49036)
  • 10. Murry Rosenblatt.
    Remarks on a multivariate transformation.
    Ann. Math. Statist., 23(3):470-472, 1952. MR 0049525 (14:189j)
  • 11. Jean Taylor.
    The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces.
    Annals of Math., 103:489-539, 1976. MR 0428181 (55:1208a)

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Additional Information

Rebecca Dorff
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84604
Email: beccadorff@gmail.com

Drew Johnson
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84604
Email: werd2.718@gmail.com

Gary R. Lawlor
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84604
Email: lawlor@mathed.byu.edu

Donald Sampson
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84604
Email: sampson.dcs@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2011-10872-4
Keywords: Metacalibration, calibration, isoperimetric, equitent
Received by editor(s): October 8, 2010
Received by editor(s) in revised form: October 20, 2010
Published electronically: April 26, 2011
Communicated by: Jianguo Cao
Article copyright: © Copyright 2011 American Mathematical Society

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