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Regularity of Minimal Surfaces

  • Ulrich Dierkes
  • Stefan Hildebrandt
  • Anthony J. Tromba

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 340)

Table of contents

  1. Front Matter
    Pages I-XVII
  2. Boundary Behaviour of Minimal Surfaces

    1. Front Matter
      Pages 1-1
    2. Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 3-73
    3. Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 75-212
    4. Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 213-276
  3. Geometric Properties of Minimal Surfaces and H-Surfaces

    1. Front Matter
      Pages 277-277
    2. Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 279-439
    3. Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 441-485
    4. Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 487-560
  4. Back Matter
    Pages 561-623

About this book

Introduction

Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau´s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau´s problem have no interior branch points.

Keywords

Boundary value problem Minimal surface Riemannian manifold calculus of variations conformal mappings differential geometry manifold minimal surfaces minimum regularity theory

Authors and affiliations

  • Ulrich Dierkes
    • 1
  • Stefan Hildebrandt
    • 2
  • Anthony J. Tromba
    • 3
  1. 1.Faculty of MathematicsUniversity of DuisburgDuisburgGermany
  2. 2.Mathematical InstituteUniversity of BonnBonnGermany
  3. 3.Baskin 621B, Department of MathematicsUniversity of California at Santa CruzSanta CruzUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-11700-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 2010
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-11699-5
  • Online ISBN 978-3-642-11700-8
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site