Authors:
- 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
- Includes supplementary material: sn.pub/extras
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (9 chapters)
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Front Matter
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Back Matter
About this book
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.
Reviews
From the reviews:
“The author provides a self-contained study of a theory of branched minimal surfaces. … The goal of the book is to study the question whether an area minimizing surface spanning a contour in three dimensional space is immersed or not, that is does its derivative have maximal rank everywhere. The exposition starts with some simple examples and continues with nicely presented proofs. The book can be useful for a graduate course or seminar.” (Themistocles M. Rassias, Zentralblatt MATH, Vol. 1247, 2012)
Authors and Affiliations
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Department of Mathematics, University of California at Santa Cruz, Santa Cruz, USA
Anthony Tromba
Bibliographic Information
Book Title: A Theory of Branched Minimal Surfaces
Authors: Anthony Tromba
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-642-25620-2
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag Berlin Heidelberg 2012
Hardcover ISBN: 978-3-642-25619-6Published: 06 January 2012
Softcover ISBN: 978-3-642-43520-1Published: 22 February 2014
eBook ISBN: 978-3-642-25620-2Published: 05 January 2012
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: X, 194
Topics: Functions of a Complex Variable, Sequences, Series, Summability, Differential Geometry, Global Analysis and Analysis on Manifolds