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Global Analysis of Minimal Surfaces

  • Book
  • © 1992

Overview

Authors:
  1. Ulrich Dierkes

    1. Faculty of Mathematics, University of Duisburg, Duisburg, Germany

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  2. Stefan Hildebrandt

    1. Mathematical Institute, University of Bonn, Bonn, Germany

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  3. Anthony J. Tromba

    1. Baskin 621B, Department of Mathematics, University of California at Santa Cruz, Santa Cruz, USA

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  • Together with vol.
  • 340 it is the long expected 2nd edition of the Grundlehren vol.
  • First part is the extension of the results treated in volumes 339 and 340 Second Part contains a "global theory of minimal surfaces" as envisioned by Smale
  • Includes supplementary material: sn.pub/extras

Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 341)

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Table of contents (6 chapters)

  1. Front Matter

    Pages I-XVI
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  2. Free Boundaries and Bernstein Theorems

    1. Front Matter

      Pages 1-1
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    2. Minimal Surfaces with Supporting Half-Planes

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 3-35

    3. Embedded Minimal Surfaces with Partially Free Boundaries

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 37-133

    4. Bernstein Theorems and Related Results

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 135-246

  3. Global Analysis of Minimal Surfaces

    1. Front Matter

      Pages 247-247
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    2. The General Problem of Plateau: Another Approach

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 249-297

    3. The Index Theorems for Minimal Surfaces of Zero and Higher Genus

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 299-400

    4. Euler Characteristic and Morse Theory for Minimal Surfaces

      • Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
      Pages 401-475

  4. Back Matter

    Pages 477-537
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Keywords

About this book

Many properties of minimal surfaces are of a global nature, and this is already true for the results treated in the first two volumes of the treatise. Part I of the present book can be viewed as an extension of these results. For instance, the first two chapters deal with existence, regularity and uniqueness theorems for minimal surfaces with partially free boundaries. Here one of the main features is the possibility of "edge-crawling" along free parts of the boundary. The third chapter deals with a priori estimates for minimal surfaces in higher dimensions and for minimizers of singular integrals related to the area functional. In particular, far reaching Bernstein theorems are derived. The second part of the book contains what one might justly call a "global theory of minimal surfaces" as envisioned by Smale. First, the Douglas problem is treated anew by using Teichmüller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to Plateau´s problem are discussed. Finally, a topological approach to minimal surfaces via Fredholm vector fields in the spirit of Smale is presented.

Reviews

From the reviews of the second edition:

“The most complete and thorough record of the ongoing efforts to justify Lagrange’s optimism. … contain a wealth of new material in the form of newly written chapters and sections … . a compilation of results and proofs from a vast subject. Here were true scholars in the best sense of the word at work, creating their literary lifetime achievements. They wrote with love for detail, clarity and history, which makes them a pleasure to read. … will become instantaneous classics.” (Matthias Weber, The Mathematical Association of America, June, 2011)

Authors and Affiliations

  • Faculty of Mathematics, University of Duisburg, Duisburg, Germany

    Ulrich Dierkes

  • Mathematical Institute, University of Bonn, Bonn, Germany

    Stefan Hildebrandt

  • Baskin 621B, Department of Mathematics, University of California at Santa Cruz, Santa Cruz, USA

    Anthony J. Tromba

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