Singular Boundary Points of Minimal Surfaces

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


The first section is devoted to the study of minimal surfaces in the neighbourhood of boundary branch points. The fundamental tool for dealing with this problem is the method of Hartman–Wintner which yields asymptotic expansions for complex-valued solutions f(w) of a differential inequality
$$|f_{\overline{w}}(w)|\leq c|w|^{-\lambda}|f(w)|\quad \mbox{on}\ B_{R}(0)$$
at the center w=0 of a disk B R (0)={w∈ℂ:|w|<R}, and more general, of expansions for vector-valued solutions X(w) of a differential inequality
$$|\Delta X(w)|\le c|w|^{-\lambda}\{|X(w)|+|\nabla X(w)|\}\quad \mbox{on}\ B_R(0)\ \mbox{at}\ w=0.$$
Such expansions are used in the preceding chapter. The remainder of the chapter deals with results by G. Dziuk investigating minimal surfaces at boundary points which are mapped onto vertices of fixed or free boundaries.


Asymptotic Expansion Minimal Surface Nonnegative Integer Jordan Curve Free Boundary Problem 
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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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