Format:
Online-Ressource (XVI, 708p. 140 illus, digital)
ISBN:
9783642116988
Series Statement:
Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 339
Content:
to the Geometry of Surfaces and to Minimal Surfaces -- Differential Geometry of Surfaces in Three-Dimensional Euclidean Space -- Minimal Surfaces -- Representation Formulas and Examples of Minimal Surfaces -- Plateau's Problem -- The Plateau Problem and the Partially Free Boundary Problem -- Stable Minimal- and H-Surfaces -- Unstable Minimal Surfaces -- Graphs with Prescribed Mean Curvature -- to the Douglas Problem -- Problems.
Content:
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 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.
Note:
Description based upon print version of record
,
Preface; Contents; Introduction; Part I. Introduction to the Geometry of Surfaces and to Minimal Surfaces; Differential Geometry of Surfaces in Three-Dimensional Euclidean Space; Minimal Surfaces; Representation Formulas and Examples of Minimal Surfaces; Part II. Plateau's Problem; The Plateau Problem and the Partially Free Boundary Problem; Stable Minimal- and H-Surfaces; Unstable Minimal Surfaces; Graphs with Prescribed Mean Curvature; Introduction to the Douglas Problem; Problems; On Relative Minimizers of Area and Energy; Minimal Surfaces in Heisenberg Groups; Bibliography; Index;
Additional Edition:
ISBN 9783642116971
Additional Edition:
Buchausg. u.d.T. Minimal surfaces ; 1: Minimal surfaces Berlin : Springer, 2010 ISBN 9783642116971
Additional Edition:
Erscheint auch als Druck-Ausgabe Dierkes, Ulrich, 1956 - [Minimal surfaces] ; 1: Minimal surfaces Heidelberg [u.a.] : Springer, 2010 ISBN 9783642116971
Language:
English
Subjects:
Mathematics
Keywords:
Minimalfläche
;
Minimalfläche
DOI:
10.1007/978-3-642-11698-8
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
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Author information:
Hildebrandt, Stefan 1936-2015
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