Minimal Surfaces
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 begin...
Main Authors:  , , 

Format:  eBook 
Language:  English 
Published: 
Berlin, Heidelberg
Springer Berlin Heidelberg
2010, 2010

Edition:  2nd ed. 2010 
Series:  Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics

Subjects:  
Online Access:  
Collection:  Springer eBooks 2005  Collection details see MPG.ReNa 
Summary:  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 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. 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. of stable surfaces of prescribed mean curvature H), especially of cmcsurfaces (H = const), and leads to curvature estimates for stable, immersed cmcsurfaces 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 Hsurfaces is solved, using the solution of Plateau´s problem for Hsurfaces and the pertinent estimates 

Physical Description:  XVI, 692 p. 149 illus., 9 illus. in color online resource 
ISBN:  9783642116988 