manuscripta mathematica

, 131:179 | Cite as

On the negativity of higher order derivatives of Dirichlet’s energy in plateau’s problem

Open Access
Article

Abstract

We calculate higher order derivatives of Dirichlet’s Energy at a branched minimal surface in the direction of Forced Jacobi Fields discovered by the author and R. Böhme. We show that, under certain conditions these derivatives can be made negative, while all lower order derivatives vanish. This is the first time that derivatives of order greater than three have been calculated.

Mathematics Subject Classification (2000)

49Q05 58E12 

References

  1. 1.
    Alt H.W.: Verzweigungspunkte von H-Flächen. I. Math. Z. 127, 333–362 (1972)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alt H.W.: Verweigungspunkte von H-Flächen. II. Math. Ann. 201, 33–55 (1973)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alt H.W., Tomi F.: Regularity and finiteness of solutions to the free boundary problem for minimal surfaces. Math. Z. 189, 227–237 (1985)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Beeson M.: On interior branch points of minimal surfaces. Math. Z. 171, 133–154 (1980)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Böhme R., Tromba A.: The index theorem for classical minimal surfaces. Ann. Math. 113, 447–499 (1981)CrossRefGoogle Scholar
  6. 6.
    Courant R.: On a generalized form of Plateau’s problem. Trans. Am. Math. Soc. 50, 40–47 (1941)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Douglas J.: One-sided minimal surfaces with a given boundary. Trans. Am. Math. Soc. 34, 731–756 (1932)CrossRefGoogle Scholar
  8. 8.
    Gulliver R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. Math. 97, 275–305 (1973)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Gulliver R., Lesley F.D.: On boundary branch points of minimizing surfaces. Arch. Ration. Mech. Anal. 52, 20–25 (1973)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gulliver R., Osserman R., Royden H.L.: A theory of branched immersions of surfaces. Am. J. Math. 95, 750–812 (1973)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Micallef M., White B.: The structure of branch points in minimal surfaces and in pseudoholomorphic curves. Ann. Math. 141, 35–85 (1995)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Osserman, R.: A proof of the regularity everywhere of the classical solution of Plateau’s problem. Ann. Math. (2) 91, 550–569 (1970)Google Scholar
  13. 13.
    Radó T.: On the problem of Plateau, Ergebnisse der Math. Band 2. Springer, Berlin (1933)Google Scholar
  14. 14.
    Tromba A.J.: Intrinsic third derivatives for Plateau’s problem and the Morse inequalities for disc minimal surfaces in \({\mathbb{R}^3}\) . Calc. Var 1, 335–353 (1993)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Tromba, A.J.: On interior branch points for minimal surfaces; to appear in: Annales de l’Institut Henri Poincaré, Analyse Non-LineaireGoogle Scholar
  16. 16.
    Wienholtz, D.: Der Ausschluß von eigentlichen Verzweigungspunkten bei Minimalflächen vom Typ der Kreisscheibe. Vorlesungsreihe 37, Sonderforschungsbereich 256, Bonn (1996)Google Scholar
  17. 17.
    Wienholtz D.: A method to exclude branch points of minimal surfaces. Calc. Var. 7, 219–247 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.University of CaliforniaSanta CruzUSA

Personalised recommendations