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Regularity of Minimal Surfaces

Regularity of Minimal Surfaces
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Field name Details
Dewey Class 515.64
Title Regularity of Minimal Surfaces (EB) / by Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba.
Author Dierkes, Ulrich
Added Personal Name Hildebrandt, Stefan , 1936-
Tromba, Anthony J. , 1943-
Other name(s) SpringerLink (Online service)
Edition statement Revised and enlarged 2nd edition
Publication Berlin, Heidelberg : Springer , 2010.
Physical Details XVII, 623 pages : online resource.
Series Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 0072-7830 ; ; 340
ISBN 9783642117008
Summary Note 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 Plateau´s 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 Plateau´s problem have no interior branch points.:
Contents note Introduction -- Part I. Boundary Behaviour of Minimal Surfaces -- 1.Minimal Surfaces with Free Boundaries -- 2.The Boundary Behaviour of Minimal -- 3.Singular Boundary Points of Minimal Surfaces -- Part II. Geometric Properties of Minimal Surfaces -- 4.Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities -- 5.The Thread Problem -- 6.Branch Points -- Bibliography -- Index.
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