MINIMAL SURFACES by Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny ( 9783642116971 )

Item Details

Title:

MINIMAL SURFACES

By:

Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny

Format:

Hardback

List price:	£109.99
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ISBN 10:

3642116973

ISBN 13:

9783642116971

Publisher:

SPRINGER-VERLAG BERLIN AND HEIDELBERG GMBH & CO. KG

Pub. date:

29 September, 2010

Edition:

2nd, rev. and enlarged ed. 2010

Series:

Grundlehren der mathematischen Wissenschaften 339

Pages:

692

Description:

This is the first of a three-volume treatise on minimal surfaces. It covers the classical theory as well as existence results concerning boundary value problems for minimal surfaces, in particular results for Plateau's problem.

Synopsis:

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 Bjorling'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 and Tomi'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.

Illustrations:

9 Illustrations, color; 140 Illustrations, black and white; XVI, 692 p. 149

Publication:

Germany

Imprint:

Springer-Verlag Berlin and Heidelberg GmbH & Co. K

Returns:

Returnable

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