Minimal Surfaces

Author:   Ulrich Dierkes ,  Ruben Jakob ,  Stefan Hildebrandt ,  Albrecht Küster
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Edition:   2nd, rev. and enlarged ed. 2010
Volume:   339
ISBN:  

9783642116971


Pages:   692
Publication Date:   01 October 2010
Format:   Hardback
Availability:   In Print   Availability explained
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Minimal Surfaces


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Overview

Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces. The treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296). The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X: \Omega\to\R^3 which is conformally parametrized on \Omega\subset\R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Björling´s initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The second part of this volume begins with a survey of Plateau´s problem and of some of its modifications. One of the main features is a new, completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal surfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche´s uniqueness theorem andTomi´s finiteness result. In addition, a theory of unstable solutions of Plateau´s problems is developed which is based on Courant´s mountain pass lemma. Furthermore, Dirichlet´s problem for nonparametric H-surfaces is solved, using the solution of Plateau´s problem for H-surfaces and the pertinent estimates.

Full Product Details

Author:   Ulrich Dierkes ,  Ruben Jakob ,  Stefan Hildebrandt ,  Albrecht Küster
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2nd, rev. and enlarged ed. 2010
Volume:   339
Dimensions:   Width: 15.50cm , Height: 3.80cm , Length: 23.50cm
Weight:   1.220kg
ISBN:  

9783642116971


ISBN 10:   3642116973
Pages:   692
Publication Date:   01 October 2010
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print   Availability explained
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

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Reviews

From the reviews of the second edition: ""This volume is in many ways an introduction to differential geometry and to the classical theory of minimal surfaces, and the first four chapters should be readable for graduate students since the only prerequisites are the elements of vector analysis and some basic knowledge of complex analysis. ... In general, the material of this volume is self-contained ... . For further study the authors refer to the extensive bibliography as well as to comments and references in the Scholia attached to each chapter."" (Andrew Bucki, Mathematical Reviews, Issue 2012 b)


From the reviews of the second edition: This volume is in many ways an introduction to differential geometry and to the classical theory of minimal surfaces, and the first four chapters should be readable for graduate students since the only prerequisites are the elements of vector analysis and some basic knowledge of complex analysis. ... In general, the material of this volume is self-contained ... . For further study the authors refer to the extensive bibliography as well as to comments and references in the Scholia attached to each chapter. (Andrew Bucki, Mathematical Reviews, Issue 2012 b)


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