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.
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Alt H.W.: Verzweigungspunkte von H-Flächen. I. Math. Z. 127, 333–362 (1972)
Alt H.W.: Verweigungspunkte von H-Flächen. II. Math. Ann. 201, 33–55 (1973)
Alt H.W., Tomi F.: Regularity and finiteness of solutions to the free boundary problem for minimal surfaces. Math. Z. 189, 227–237 (1985)
Beeson M.: On interior branch points of minimal surfaces. Math. Z. 171, 133–154 (1980)
Böhme R., Tromba A.: The index theorem for classical minimal surfaces. Ann. Math. 113, 447–499 (1981)
Courant R.: On a generalized form of Plateau’s problem. Trans. Am. Math. Soc. 50, 40–47 (1941)
Douglas J.: One-sided minimal surfaces with a given boundary. Trans. Am. Math. Soc. 34, 731–756 (1932)
Gulliver R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. Math. 97, 275–305 (1973)
Gulliver R., Lesley F.D.: On boundary branch points of minimizing surfaces. Arch. Ration. Mech. Anal. 52, 20–25 (1973)
Gulliver R., Osserman R., Royden H.L.: A theory of branched immersions of surfaces. Am. J. Math. 95, 750–812 (1973)
Micallef M., White B.: The structure of branch points in minimal surfaces and in pseudoholomorphic curves. Ann. Math. 141, 35–85 (1995)
Osserman, R.: A proof of the regularity everywhere of the classical solution of Plateau’s problem. Ann. Math. (2) 91, 550–569 (1970)
Radó T.: On the problem of Plateau, Ergebnisse der Math. Band 2. Springer, Berlin (1933)
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)
Tromba, A.J.: On interior branch points for minimal surfaces; to appear in: Annales de l’Institut Henri Poincaré, Analyse Non-Lineaire
Wienholtz, D.: Der Ausschluß von eigentlichen Verzweigungspunkten bei Minimalflächen vom Typ der Kreisscheibe. Vorlesungsreihe 37, Sonderforschungsbereich 256, Bonn (1996)
Wienholtz D.: A method to exclude branch points of minimal surfaces. Calc. Var. 7, 219–247 (1998)
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Tromba, A.J. On the negativity of higher order derivatives of Dirichlet’s energy in plateau’s problem. manuscripta math. 131, 179–197 (2010). https://doi.org/10.1007/s00229-009-0316-x
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DOI: https://doi.org/10.1007/s00229-009-0316-x