Minimal Surfaces Paperback / softback
by Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny
Part of the Grundlehren der mathematischen Wissenschaften series
Paperback / softback
Description
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.
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Item not Available
- Format:Paperback / softback
- Pages:692 pages, 9 Illustrations, color; 140 Illustrations, black and white; XVI, 692 p. 149 illus., 9 ill
- Publisher:Springer-Verlag Berlin and Heidelberg GmbH & Co. K
- Publication Date:01/12/2012
- Category:
- ISBN:9783642265273
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Information
-
Item not Available
- Format:Paperback / softback
- Pages:692 pages, 9 Illustrations, color; 140 Illustrations, black and white; XVI, 692 p. 149 illus., 9 ill
- Publisher:Springer-Verlag Berlin and Heidelberg GmbH & Co. K
- Publication Date:01/12/2012
- Category:
- ISBN:9783642265273
