Grundlehren der mathematischen Wissenschaften

Global Analysis of Minimal Surfaces

Authors: Dierkes, Ulrich, Hildebrandt, Stefan, Tromba, Anthony

  • Together with vol. 340 it is the long expected 2nd edition of the Grundlehren vol. 296 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

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eBook $89.00
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  • ISBN 978-3-642-11706-0
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Hardcover $119.00
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  • ISBN 978-3-642-11705-3
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Softcover $119.00
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  • ISBN 978-3-642-26533-4
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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)


Table of contents (6 chapters)

  • Minimal Surfaces with Supporting Half-Planes

    Dierkes, Ulrich (et al.)

    Pages 3-35

  • Embedded Minimal Surfaces with Partially Free Boundaries

    Dierkes, Ulrich (et al.)

    Pages 37-133

  • Bernstein Theorems and Related Results

    Dierkes, Ulrich (et al.)

    Pages 135-246

  • The General Problem of Plateau: Another Approach

    Dierkes, Ulrich (et al.)

    Pages 249-297

  • The Index Theorems for Minimal Surfaces of Zero and Higher Genus

    Dierkes, Ulrich (et al.)

    Pages 299-400

Buy this book

eBook $89.00
price for USA (gross)
  • ISBN 978-3-642-11706-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $119.00
price for USA
  • ISBN 978-3-642-11705-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $119.00
price for USA
  • ISBN 978-3-642-26533-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Global Analysis of Minimal Surfaces
Authors
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
341
Copyright
1992
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-11706-0
DOI
10.1007/978-3-642-11706-0
Hardcover ISBN
978-3-642-11705-3
Softcover ISBN
978-3-642-26533-4
Series ISSN
0072-7830
Edition Number
2
Number of Pages
XVI, 537
Number of Illustrations and Tables
41 b/w illustrations, 5 illustrations in colour
Additional Information
Originally published as part of volume 296 in the series: Grundlehren der mathematischen Wissenschaften
Topics