Minimal Surfaces

Ulrich Dierkes,Stefan Hildebrandt,Friedrich Sauvigny

Minimal Surfaces
Format
Paperback
Publisher
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Country
Germany
Published
1 December 2012
Pages
692
ISBN
9783642265273

Minimal Surfaces

Ulrich Dierkes,Stefan Hildebrandt,Friedrich Sauvigny

This title is printed to order. This book may have been self-published. If so, we cannot guarantee the quality of the content. In the main most books will have gone through the editing process however some may not. We therefore suggest that you be aware of this before ordering this book. If in doubt check either the author or publisher’s details as we are unable to accept any returns unless they are faulty. Please contact us if you have any questions.

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 Bjoerlings 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 Plateaus 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 Nitsches uniqueness theorem and Tomis finiteness result. In addition, a theory of unstable solutions of Plateaus problems is developed which is based on Courants mountain pass lemma. Furthermore, Dirichlets problem for nonparametric H-surfaces is solved, using the solution of Plateaus problem for H-surfaces and the pertinent estimates.

This item is not currently in-stock. It can be ordered online and is expected to ship in 7-14 days

Our stock data is updated periodically, and availability may change throughout the day for in-demand items. Please call the relevant shop for the most current stock information. Prices are subject to change without notice.

Sign in or become a Readings Member to add this title to a wishlist.