Journal of Geometric Analysis

, Volume 24, Issue 1, pp 271–297 | Cite as

Soap Film Solutions to Plateau’s Problem

Article

Abstract

Plateau’s problem is to show the existence of an area-minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area-minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau’s problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator, a Poincaré Lemma, and good compactness properties to solve Plateau’s problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Möbius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem. We use the chain complex of differential chains, a geometric Poincaré Lemma, and good compactness properties of the complex to solve Plateau’s problem in such generality as to find the first solution which minimizes area taken from a collection of surfaces that includes all previous special cases, as well as all smoothly immersed surfaces of any genus type, orientable or nonorientable, and surfaces with multiple junctions. Our result holds for all dimensions and codimension-one surfaces in ℝn.

Keywords

Plateau’s problem Differential chains Differential forms Chainlets Dipole chains Dirac chains Poincare Lemma Extrusion Retraction Prederivative Pushforward Volume functional Soap films Triple branches Moebius strips Compactness Minimal sets 

Mathematics Subject Classification

49Q15 49J52 49J99 

References

  1. 1.
    Almgren, F.J.: Plateau’s Problem, an Invitation to Varifold Geometry. Benjamin, Elmsford (1966) MATHGoogle Scholar
  2. 2.
    Almgren, F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Bull. Am. Math. Soc. 81(1), 151–154 (1975) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Alt, H.W.: Verzweigungspunkte von H-Flächen. ii. Math. Ann. 201, 33–55 (1973) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Douglas, J.: Solutions of the problem of Plateau. Trans. Am. Math. Soc. 33, 263–321 (1931) CrossRefGoogle Scholar
  5. 5.
    Federer, H.: Geometric Measure Theory. Springer, Berlin (1969) MATHGoogle Scholar
  6. 6.
    Federer, H., Fleming, W.H.: Normal and integral currents. Ann. Math. 72(3), 458–520 (1960) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Fleming, W.H.: On the oriented Plateau problem. Rend. Circ. Mat. Palermo 11(1), 69–90 (1962) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Fleming, W.H.: Flat chains over a finite coefficient group. Trans. Am. Math. Soc. 121(1), 160–186 (1966) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. Math. 97(2), 275–305 (1973) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Harrison, J.: Cartan’s magic formula and soap film structures. J. Geom. Anal. 14(1), 47–61 (2004) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Harrison, J.: On Plateau’s problem for soap films with a bound on energy. J. Geom. Anal. 14(2), 319–329 (2004) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Harrison, J.: Operator calculus of differential chains and differential forms. J. Geom. Anal., to appear Google Scholar
  13. 13.
    Harrison, J.: Differential chains, measures, and additive set functions (July 2012) Google Scholar
  14. 14.
    Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Bull. Am. Math. Soc. 1(1), 263–265 (1979) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Morgan, F.: Geometric Measure Theory: A Beginners Guide. Academic Press, London (1988) Google Scholar
  16. 16.
    Osserman, R.: A proof of the regularity everywhere of the classical solution to Plateau’s problem. Ann. Math. 91, 550–569 (1970) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Plateau, J.: Experimental and Theoretical Statics of Liquids Subject to Molecular Forces Only. Gauthier-Villars, Paris (1873) Google Scholar
  18. 18.
    Reifenberg, E.R.: Solution of the Plateau problem for m-dimensional surfaces of varying topological type. Acta Math. 80(2), 1–14 (1960) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957) MATHGoogle Scholar
  20. 20.
    Ziemer, W.P.: Integral currents mod 2. Trans. Am. Math. Soc. 105, 496–524 (1962) MATHMathSciNetGoogle Scholar
  21. 21.
    Ziemer, W.P.: Plateau’s problem: an invitation to varifold geometry. Bull. Am. Math. Soc. 75(5), 924–925 (1969) CrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations