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The isoperimetric problem in higher codimension

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

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References

  1. Allard W.K.: On the first variation of a varifold. Ann. Math. 95, 417–491 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allard W.K.: On boundary regularity for Plateau’s problem. Bull. Am. Math. Soc. 75, 522–523 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  3. Almgren F.: Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35, 451–547 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  5. Barbosa J.L., do Carmo M.: Stability of hypersurfaces with constant mean curvature. Math. Z. 185, 339–353 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bombieri E., De Giorgi E., Giusti E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dierkes U., Hildebrandt S., Kuster A., Wohlrab O.: Minimal Surfaces II: Boundary Regularity. Springer, New York (1992)

    Book  MATH  Google Scholar 

  8. Duzaar F., Fuchs M.: On the existence of integral currents with prescribed mean curvature vector. Manuscripta Math. 67, 41–67 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duzaar F., Fuchs M.: On integral currents with constant mean curvature. Rend. Sem. Univ. Padova 85, 79–103 (1991)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  11. Duzaar F., Steffen K.: λ Minimizing currents. Manuscripta Math. 80, 403–407 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ecker K.: Area-minimizing integral currents with movable boundary parts of prescribed mass. Ann. Inst. H. Poincare 6, 261–293 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Federer H.: Real flat chains, cochains, and variational problems. Indiana Univ. Math. J. 24, 351–407 (1974/75)

    Article  MathSciNet  Google Scholar 

  14. Gulliver R.: Existence of surfaces with prescribed mean curvature vector. Math. Z. 131, 117–140 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  16. Lawlor G.: A sufficient condition for a cone to be area-minimizing. Mem. Am. Math. Soc. 91, 446 (1991)

    MathSciNet  Google Scholar 

  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)

  18. Morgan F.: Geometric Measure Theory: a Beginner’s Guide. Academic Press, San Diego (2009)

    Google Scholar 

  19. Morgan F.: An isoperimetric inequality for the thread problem. Bull. Austral. Math. Soc. 55, 489–495 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Morgan F.: Measures on spaces of surfaces. Arch. Rat. Mech. Anal. 78, 335–359 (1982)

    Article  MATH  Google Scholar 

  21. Morgan F.: Perimeter-minimizing curves and surfaces in \({\mathbf{R}^n}\) enclosing prescribed multi-volume. Asian J. Math. 4, 373–382 (2000)

    MathSciNet  MATH  Google Scholar 

  22. Morgan F.: Strict calibrations. Matemática Contemp. 9, 139–152 (1995)

    MATH  Google Scholar 

  23. Nitsche, J.C.C.: Vorlesungen über Minimalflächen. Springer, New York, 1975; Translation: Lectures on Minimal Surfaces. Cambridge University Press, New York (1989)

  24. Salavessa I.M.C.: Stability of submanifolds with parallel mean curvature in calibrated manifolds. Bull. Braz. Math. Soc. 41, 495–530 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Salavessa I.M.C.: Stable 3-spheres in \({\mathbf{C}^3}\). J. Math. Res. 4(2), 34–44 (2012)

    MathSciNet  Google Scholar 

  26. Schoen, R.: Minimal submanifolds in higher codimension. Matemática Contemporânea 30, 169–199 (2006). http://www.mat.unb.br/~matcont/30_11.pdf

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Bronfman Science Center, Williams College, 18 Hoxsey Street, Williamstown, MA, 01267, USA

    Frank Morgan

  2. Centro de Física das Interacções Fundamentais, Instituto Superior Técnico, Technical University of Lisbon, Edifício Ciência Piso 3, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Isabel M. C. Salavessa

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  1. Frank Morgan
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  2. Isabel M. C. Salavessa
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Correspondence to Isabel M. C. Salavessa.

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Morgan, F., Salavessa, I.M.C. The isoperimetric problem in higher codimension. manuscripta math. 142, 369–382 (2013). https://doi.org/10.1007/s00229-012-0604-8

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  • DOI: https://doi.org/10.1007/s00229-012-0604-8

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