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AuthorDierkes, Ulrich. author
TitleGlobal Analysis of Minimal Surfaces [electronic resource] / by Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1992
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Descript XVI, 537 p. 46 illus., 5 illus. in color. online resource


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 Teichmรผller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to Plateauยดs problem are discussed. Finally, a topological approach to minimal surfaces via Fredholm vector fields in the spirit of Smale is presented


Introduction -- Part I. Free Boundaries and Bernstein Theorems -- 1.Minimal Surfaces with Supporting Half-Planes -- 2.Embedded Minimal Surfaces with Partially Free Boundaries -- 3.Bernstein Theorems and Related Results -- Part II. Global Analysis of Minimal Surfaces -- 4.The General Problem of Plateau: Another Approach -- 5.The Index Theorems for Minimal Surfaces of Zero and Higher Genus -- 6.Euler Characteristic and Morse Theory for Minimal Surfaces -- Bibliography -- Index

Mathematics Functions of complex variables Global analysis (Mathematics) Manifolds (Mathematics) Partial differential equations Differential geometry Calculus of variations Physics Mathematics Calculus of Variations and Optimal Control; Optimization Differential Geometry Partial Differential Equations Functions of a Complex Variable Theoretical Mathematical and Computational Physics Global Analysis and Analysis on Manifolds


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