Grundlehren der mathematischen Wissenschaften

Regularity of Minimal Surfaces

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

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  • -together with vol. 341 it is the expected 2nd edition of the Grundlehren vol. 296 -discusses geometric properties of minimal and H-surfaces -includes a new approach to the Osserman-Gulliver-Alt theorem

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About this book

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.

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)

Table of contents (6 chapters)

Buy this book

eBook $99.00
price for USA in USD (gross)
  • ISBN 978-3-642-11700-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.99
price for USA in USD
  • ISBN 978-3-642-11699-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $129.00
price for USA in USD
  • ISBN 978-3-642-26521-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Regularity of Minimal Surfaces
Authors
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
340
Copyright
2010
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-11700-8
DOI
10.1007/978-3-642-11700-8
Hardcover ISBN
978-3-642-11699-5
Softcover ISBN
978-3-642-26521-1
Series ISSN
0072-7830
Edition Number
2
Number of Pages
XVII, 623
Number of Illustrations
62 b/w illustrations, 6 illustrations in colour
Additional Information
Originally published as part of volume 296 in the series: Grundlehren der mathematischen Wissenschaft
Topics