Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities

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


This is the second central chapter of Vol. 2, collecting a wealth of information. It could have been labeled as geometric properties of minimal surfaces. First we derive inclusion theorems for minimal surfaces in dependence of their boundary data. Such results are more or less sophisticated versions of the maximum principle. They lead to interesting nonexistence results for connected minimal surfaces and H-surfaces whose boundaries consist of several disjoint components. Here even the situation for higher dimensional surfaces and for solutions of variational inequalities, obtained from obstacle problems, is discussed. Inclusion principles for such solutions are the fundament for results ensuring the existence of minimal surfaces and H-surfaces solving Plateau’s problem in Euclidean space or in a Riemannian manifold respectively. Of particular interest in this context are Jacobi field estimates.

In addition, isoperimetric inequalities for minimal surfaces solving either Plateau’s problem or a free boundary problem are derived.


Minimal Surface Principal Curvature Isoperimetric Inequality Curvature Vector Obstacle 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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