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A Theory of Branched Minimal Surfaces

  • Kartonierter Einband
  • 204 Seiten
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This book shows how to calculate arbitrarily high orders of derivatives of the Douglas Energy defined on the infinite dimensional ... Weiterlesen
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Beschreibung

This book shows how to calculate arbitrarily high orders of derivatives of the Douglas Energy defined on the infinite dimensional manifold of all surfaces spanning a contour, breaking new ground in the Calculus of Variations.


One 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 the second derivative or variation is calculated. The monograph begins with easy examples leading to a proof in a large number of cases that can be presented in a graduate course in either manifolds or complex analysis. Thus this monograph requires only the most basic knowledge of analysis, complex analysis and topology and can therefore be read by almost anyone with a basic graduate education.

Presents an elementary proof of a very fundamental and beautiful mathematical result

First complete presentation of this results in the mathematical literature

It can be read by almost anyone with a basic graduate education



Inhalt
1.Introduction.- 2.Higher order Derivatives of Dirichlets' Energy.- 3.Very Special Case; The Theorem for n + 1 Even and m + 1 Odd .- 4.The First Main Theorem; Non-Exceptional Branch Points.- 5.The Second Main Theorem: Exceptional Branch Points; The Condition k > l.- 6.Exceptional Branch Points Without The Condition k > l.- 7.New Brief Proofs of the Gulliver-Osserman-Royden Theorem .- 8.Boundary Branch Points.- Scholia.- Appendix.- Bibliography.

Produktinformationen

Titel: A Theory of Branched Minimal Surfaces
Autor:
EAN: 9783642435201
ISBN: 3642435203
Format: Kartonierter Einband
Herausgeber: Springer Berlin Heidelberg
Genre: Mathematik
Anzahl Seiten: 204
Gewicht: 318g
Größe: H235mm x B155mm x T11mm
Jahr: 2014
Auflage: 2012

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