Manuscripta Mathematica

, Volume 142, Issue 3–4, pp 369–382 | Cite as

The isoperimetric problem in higher codimension

  • Frank Morgan
  • Isabel M. C. SalavessaEmail author


We consider three generalizations of the isoperimetric problem to higher codimension and provide results on equilibrium, stability, and minimization.

Mathematics Subject Classification (2000)

Primary 49Q20 Secondary 53A10 49Q10 53C42 


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  1. 1.
    Allard W.K.: On the first variation of a varifold. Ann. Math. 95, 417–491 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allard W.K.: On boundary regularity for Plateau’s problem. Bull. Am. Math. Soc. 75, 522–523 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Almgren F.: Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35, 451–547 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Almgren F.J. Jr.: Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two. Bull. Am. Math. Soc. 8, 327–328 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barbosa J.L., do Carmo M.: Stability of hypersurfaces with constant mean curvature. Math. Z. 185, 339–353 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bombieri E., De Giorgi E., Giusti E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dierkes U., Hildebrandt S., Kuster A., Wohlrab O.: Minimal Surfaces II: Boundary Regularity. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  8. 8.
    Duzaar F., Fuchs M.: On the existence of integral currents with prescribed mean curvature vector. Manuscripta Math. 67, 41–67 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duzaar F., Fuchs M.: On integral currents with constant mean curvature. Rend. Sem. Univ. Padova 85, 79–103 (1991)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Duzaar F., Fuchs M.: A general existence theorem for integral currents with prescribed mean curvature form. Boll. Un. Mat. Ital. B 7(6-B), 901–912 (1992)MathSciNetGoogle Scholar
  11. 11.
    Duzaar F., Steffen K.: λ Minimizing currents. Manuscripta Math. 80, 403–407 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ecker K.: Area-minimizing integral currents with movable boundary parts of prescribed mass. Ann. Inst. H. Poincare 6, 261–293 (1989)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Federer H.: Real flat chains, cochains, and variational problems. Indiana Univ. Math. J. 24, 351–407 (1974/75)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gulliver R.: Existence of surfaces with prescribed mean curvature vector. Math. Z. 131, 117–140 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gulliver, R.: Necessary conditions for submanifolds and currents with prescribed mean curvature vector. In: Enrico B. (ed.) Seminar on Minimal Submanifolds, Annals of Mathematical Studies, vol. 103, pp. 225–242. Princeton University Press, Princeton (1983)Google Scholar
  16. 16.
    Lawlor G.: A sufficient condition for a cone to be area-minimizing. Mem. Am. Math. Soc. 91, 446 (1991)MathSciNetGoogle Scholar
  17. 17.
    Lawson, H.B.: Compact minimal surfaces in \({\mathbf{S}^3}\). In: Global Analysis, pp. 275–282. American Mathematical Society, Providence (1970) (Proceedings of Symposia in Pure Mathematics, vol. XV. Berkeley, California, 1968)Google Scholar
  18. 18.
    Morgan F.: Geometric Measure Theory: a Beginner’s Guide. Academic Press, San Diego (2009)Google Scholar
  19. 19.
    Morgan F.: An isoperimetric inequality for the thread problem. Bull. Austral. Math. Soc. 55, 489–495 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Morgan F.: Measures on spaces of surfaces. Arch. Rat. Mech. Anal. 78, 335–359 (1982)CrossRefzbMATHGoogle Scholar
  21. 21.
    Morgan F.: Perimeter-minimizing curves and surfaces in \({\mathbf{R}^n}\) enclosing prescribed multi-volume. Asian J. Math. 4, 373–382 (2000)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Morgan F.: Strict calibrations. Matemática Contemp. 9, 139–152 (1995)zbMATHGoogle Scholar
  23. 23.
    Nitsche, J.C.C.: Vorlesungen über Minimalflächen. Springer, New York, 1975; Translation: Lectures on Minimal Surfaces. Cambridge University Press, New York (1989)Google Scholar
  24. 24.
    Salavessa I.M.C.: Stability of submanifolds with parallel mean curvature in calibrated manifolds. Bull. Braz. Math. Soc. 41, 495–530 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Salavessa I.M.C.: Stable 3-spheres in \({\mathbf{C}^3}\). J. Math. Res. 4(2), 34–44 (2012)MathSciNetGoogle Scholar
  26. 26.
    Schoen, R.: Minimal submanifolds in higher codimension. Matemática Contemporânea 30, 169–199 (2006).
  27. 27.
    Wente H.C.: A note on the stability theorem of J. L. Barbosa and M. do Carmo for closed surfaces of constant mean curvature. Pac. J. Math. 147, 375–379 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    White B.: The space of m-dimensional surfaces that are stationary for a parametric elliptic functional. Indiana Univ. Math. J. 36, 567–602 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yau, S.T.: Submanifolds with constant mean curvature. I. II. Am. J. Math. 96, 346–366 (1974); ibid 97, 76–100 (1975); MR0370443 (51 #6670)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBronfman Science Center, Williams CollegeWilliamstownUSA
  2. 2.Centro de Física das Interacções Fundamentais, Instituto Superior Técnico, Technical University of LisbonLisbonPortugal

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