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A Theory of Branched Minimal Surfaces

  • Anthony Tromba

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages I-IX
  2. Anthony Tromba
    Pages 1-5
  3. Anthony Tromba
    Pages 117-168
  4. Anthony Tromba
    Pages 169-175
  5. Back Matter
    Pages 177-191

About this book

Introduction

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.

Keywords

30B10, 49J50, 49Q05, 53A10, 58E12, 58C20 Branch Points Immersed Minimal Surfaces Minimal Surfaces Plateau's Problem

Authors and affiliations

  • Anthony Tromba
    • 1
  1. 1.Department of MathematicsUniversity of California at Santa CruzSanta CruzUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-25620-2
  • Copyright Information Springer-Verlag Berlin Heidelberg 2012
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-25619-6
  • Online ISBN 978-3-642-25620-2
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site