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A Theory of Branched Minimal Surfaces
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Catalogue Information
Field name
Details
Dewey Class
515.9
Title
A Theory of Branched Minimal Surfaces ([EBook]) / by Anthony Tromba.
Author
Tromba, Anthony J. , 1943-
Other name(s)
SpringerLink (Online service)
Publication
Berlin, Heidelberg : Springer , 2012.
Physical Details
IX, 191 pages, 2 illus. : online resource.
Series
Springer monographs in mathematics
1439-7382
ISBN
9783642256202
Summary Note
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.:
Contents note
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.
System details note
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site
http://dx.doi.org/10.1007/978-3-642-25620-2
Links to Related Works
Subject References:
Differential Geometry
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Functions of a Complex Variable
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Functions of complex variables
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Global analysis
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Global Analysis and Analysis on Manifolds
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Global differential geometry
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Mathematics
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Sequences (Mathematics)
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Sequences, Series, Summability
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Authors:
Tromba, Anthony J. 1943-
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Corporate Authors:
SpringerLink (Online service)
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Series:
Springer monographs in mathematics
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Classification:
515.9
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515.9 (DDC 23)
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