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Minimal Surfacesby Ulrich Dierkes; Stefan Hildebrandt; Friedrich Sauvigny; Ruben Jakob; Albrecht Kuster
Springer 2010; US$ 129.00"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 non-constant harmonic mapping X: \Omega\to\R^3 which is conformally... more...
Regularity of Minimal Surfacesby Ulrich Dierkes; Stefan Hildebrandt; Anthony J. Tromba; Albrecht Kuster
Springer 2010; US$ 129.00"Regularity of Minimal Surfaces" begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect... more...
Global Analysis of Minimal Surfacesby Ulrich Dierkes; Stefan Hildebrandt; Anthony J. Tromba
Springer 2010; US$ 129.00Many properties of minimal surfaces are of a global nature, and this is already true for the results treated in the first two volumes of the treatise. Part I of the present book can be viewed as an extension of these results. For instance, the first two chapters deal with existence, regularity and uniqueness theorems for minimal surfaces with partially free boundaries. Here one of the main features is the possibility of 'edge-crawling' along free parts of the boundary. The third chapter deals with a priori estimates for minimal surfaces in higher dimensions and for minimizers of singular integrals related to the area functional. In particular, far reaching Bernstein theorems are derived. The second part of the book contains what one... more...
Classi caratteristiche e questioni connesseby E Martinelli
Springer 2011; US$ 49.95I. Bucur: L anneau de Chow d une vari t alg brique.- E. Eckmann: Cohomologie et classes caract ristiques.- C. Teleman: Sur le caract re de Chern d un fibr vectoriel complexe diff rentiable.- E. Thomas: Characteristic classes and differentiable manifolds.- A. Van de Ven: Chern classes and complex manifolds. more...
A Theory of Branched Minimal Surfacesby Anthony Tromba
Springer 2012; US$ 99.00One of the most elementary questions in mathematics is whether an area minimizing surface spanning a contour in three space is immersed or not; i.e. does its derivative have maximal rank everywhere. The purpose of this monograph is to present an elementary proof of this very fundamental and beautiful mathematical result. The exposition follows the original line of attack initiated by Jesse Douglas in his Fields medal work in 1931, namely use Dirichlet's energy as opposed to area. Remarkably, the author shows how to calculate arbitrarily high orders of derivatives of Dirichlet's energy defined on the infinite dimensional manifold of all surfaces spanning a contour, breaking new ground in the Calculus of Variations, where normally only... more...
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