Minimal Surfaces (Grundlehren der mathematischen Wissenschaften Book 339) by Ulrich Dierkes (PDF)

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: OmegaoR^3 which is conformally parametrized on OmegasubsetR^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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐

⭐

⭐

⭐

⭐

⭐

⭐

Keywords

Free Download Minimal Surfaces (Grundlehren der mathematischen Wissenschaften Book 339) in PDF format
Minimal Surfaces (Grundlehren der mathematischen Wissenschaften Book 339) PDF Free Download
Download Minimal Surfaces (Grundlehren der mathematischen Wissenschaften Book 339) 2010 PDF Free
Minimal Surfaces (Grundlehren der mathematischen Wissenschaften Book 339) 2010 PDF Free Download
Download Minimal Surfaces (Grundlehren der mathematischen Wissenschaften Book 339) PDF
Free Download Ebook Minimal Surfaces (Grundlehren der mathematischen Wissenschaften Book 339)