Dewey Class 
515.64 
Title 
Minimal Surfaces (EB) / by Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny. 
Author 
Dierkes, Ulrich 
Added Personal Name 
Hildebrandt, Stefan , 1936 
Sauvigny, Friedrich 
Other name(s) 
SpringerLink (Online service) 
Edition statement 
Revised and enlarged 2nd edition 
Publication 
Berlin, Heidelberg : Springer , 2010. 
Physical Details 
XVI, 692 pages : online resource. 
Series 
Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 00727830 ; ; 339 
ISBN 
9783642116988 
Summary Note 
Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339341). 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 threedimensional 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 BjÃ¶rlingÂ´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 PlateauÂ´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 KornLichtenstein, 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 Hsurfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmcsurfaces (H = const), and leads to curvature estimates for stable, immersed cmcsurfaces and to NitscheÂ´s uniqueness theorem and TomiÂ´s finiteness result. In addition, a theory of unstable solutions of PlateauÂ´s problems is developed which is based on CourantÂ´s mountain pass lemma. Furthermore, DirichletÂ´s problem for nonparametric Hsurfaces is solved, using the solution of PlateauÂ´s problem for Hsurfaces and the pertinent estimates.: 
Contents note 
Introduction  Part I. Introduction to the Geometry of Surfaces and to Minimal Surfaces  1.Differential Geometry of Surfaces in ThreeDimensional Euclidean Space  2.Minimal Surfaces  3.Representation Formulas and Examples of Minimal Surfaces  Part II. Plateauâs Problem  4.The Plateau Problem, and its Ramifications  5.Stable Minimal and HSurfaces  6.Unstable Minimal Surfaces  7.Graphs with Prescribed Mean Curvature  8.Introduction to the Douglas Problem  Problems  9. Appendix 1. On Relative Minimizers of Area and Energy  Appendix 2. Minimal Surfaces in Heisenberg Groups  Bibliography  Index. 
System details note 
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users). 
Internet Site 
http://dx.doi.org/10.1007/9783642116988 
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