Regularity of Minimal Surfaces: 340 (Grundlehren der mathematischen Wissenschaften) - Softcover

Hildebrandt, Stefan; Dierkes, Ulrich

 
9783642265211: Regularity of Minimal Surfaces: 340 (Grundlehren der mathematischen Wissenschaften)
Softcover
ISBN 10: 3642265219 ISBN 13: 9783642265211
Verlag: Springer, 2012

Inhaltsangabe

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.

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Von der hinteren Coverseite

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.

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Verlag
Springer
Erscheinungsdatum
2012
Sprache
Englisch
ISBN 10 
3642265219
ISBN 13 
9783642265211
Einband
Tapa blanda
Anzahl der Seiten
644
Kontakt zum Hersteller
Nicht verfügbar
Verantwortliche Person
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Vorgestellte Ausgabe

ISBN 10:  364211699X ISBN 13:  9783642116995
Verlag: Springer, 2012
Hardcover