Global Analysis of Minimal Surfaces

Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba

Global Analysis of Minimal Surfaces
Format
Hardback
Publisher
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Country
Germany
Published
4 October 2010
Pages
537
ISBN
9783642117053

Global Analysis of Minimal Surfaces

Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba

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 Teichmuller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to Plateaus 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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