Global Analysis of Minimal Surfaces (Grundlehren der mathematischen Wissenschaften, 341)

by: Ulrich Dierkes (Author),Stefan Hildebrandt(Author),Anthony Tromba(Author)&0more

Publisher: Springer

Edition: 2nd ed. 1992

Publication Date: 2010/10/4

Language: English

Print Length: 553 pages

ISBN-10: 3642117058

ISBN-13: 9783642117053

Book Description

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 Bestein 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.

About the Author

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 Bestein 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.

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