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Branch Points

  • Ulrich DierkesEmail author
  • Stefan Hildebrandt
  • Anthony J. Tromba
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 340)

Abstract

This last chapter contains a new approach to the celebrated result that a minimizer of area in a given contour has no interior branch points. The novelty of the method consists particularly in the fact that, in certain cases, relative minimizers of Dirichlet’s integral are shown to be free of nonexceptional branch points, and this is achieved by a purely analytical reasoning.

Then, boundary branch points of a minimal surface in the class \(\mathcal{C}(\Gamma)\) of admissible surfaces with a smooth boundary contour Γ are studied. In particular it is shown that a minimal surface \(\mathcal{C}(\Gamma)\) cannot be a minimizer of D in \(\mathcal{C}(\Gamma)\) if it has a boundary branch point whose order n and index m satisfy the condition 2m−2<3n (Wienholtz’s theorem).

Furthermore, geometric conditions are exhibited which furnish bounds for the index of interior and boundary branch points. These estimates supplement the bounds on the order of branch points provided by the Gauss–Bonnet theorem.

A special role in all of this is played by the forced Jacobi fields discovered by Böhme and Tromba, which will again show up in the global theory presented in Vol. 3.

Keywords

Normal Form Minimal Surface Branch Point Boundary Contour Harmonic Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Ulrich Dierkes
    • 1
    Email author
  • Stefan Hildebrandt
    • 2
  • Anthony J. Tromba
    • 3
  1. 1.Fakultät für MathematikUniversität Duisburg-Essen, Campus DuisburgDuisburgGermany
  2. 2.Mathematical InstituteUniversity of BonnBonnGermany
  3. 3.Department of MathematicsUniversity of California at Santa CruzSanta CruzUSA

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