Regularity of Minimal Surfaces

Author:   Ulrich Dierkes ,  Albrecht Kuster ,  Stefan Hildebrandt ,  Anthony Tromba
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Edition:   2nd, rev. and enlarged ed. 2010
Volume:   340
ISBN:  

9783642116995


Pages:   623
Publication Date:   30 September 2010
Format:   Hardback
Availability:   In Print   Availability explained
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Regularity of Minimal Surfaces


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Overview

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 reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateaus problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateaus problem have no interior branch points.

Full Product Details

Author:   Ulrich Dierkes ,  Albrecht Kuster ,  Stefan Hildebrandt ,  Anthony Tromba
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2nd, rev. and enlarged ed. 2010
Volume:   340
Dimensions:   Width: 15.50cm , Height: 3.40cm , Length: 23.50cm
Weight:   1.124kg
ISBN:  

9783642116995


ISBN 10:   364211699
Pages:   623
Publication Date:   30 September 2010
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print   Availability explained
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

Table of Contents

Boundary Behaviour of Minimal Surfaces.- Minimal Surfaces with Free Boundaries.- The Boundary Behaviour of Minimal Surfaces.- Singular Boundary Points of Minimal Surfaces.- Geometric Properties of Minimal Surfaces.- Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities.- The Thread Problem.- Branch Points.

Reviews

From the reviews of the second edition: The most complete and thorough record of the ongoing efforts to justify Lagrange's optimism. ... contain a wealth of new material in the form of newly written chapters and sections ... . a compilation of results and proofs from a vast subject. Here were true scholars in the best sense of the word at work, creating their literary lifetime achievements. They wrote with love for detail, clarity and history, which makes them a pleasure to read. ... will become instantaneous classics. (Matthias Weber, The Mathematical Association of America, June, 2011)


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