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Book synopsis

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 oR^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.

Book details

Book category Books in English Mathematics & science Mathematics Geometry

• Full title: Minimal Surfaces
• Author: Ulrich Dierkes
• Language: English
• Binding: Hardback
• Number of pages: 692
• EAN: 9783642116971
• ISBN: 3642116973
• ID: 04372360
• Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
• Weight: 1220 g
• Dimensions: 235 × 155 × 42 mm
• Date of publishing: 29. September 2010

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