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Complete Minimal Surfaces of Finite Total Curvature

  • Kichoon Yang

Part of the Mathematics and Its Applications book series (MAIA, volume 294)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Kichoon Yang
    Pages 1-54
  3. Kichoon Yang
    Pages 55-118
  4. Back Matter
    Pages 143-160

About this book

Introduction

This monograph contains an exposition of the theory of minimal surfaces in Euclidean space, with an emphasis on complete minimal surfaces of finite total curvature. Our exposition is based upon the philosophy that the study of finite total curvature complete minimal surfaces in R3, in large measure, coincides with the study of meromorphic functions and linear series on compact Riemann sur­ faces. This philosophy is first indicated in the fundamental theorem of Chern and Osserman: A complete minimal surface M immersed in R3 is of finite total curvature if and only if M with its induced conformal structure is conformally equivalent to a compact Riemann surface Mg punctured at a finite set E of points and the tangential Gauss map extends to a holomorphic map Mg _ P2. Thus a finite total curvature complete minimal surface in R3 gives rise to a plane algebraic curve. Let Mg denote a fixed but otherwise arbitrary compact Riemann surface of genus g. A positive integer r is called a puncture number for Mg if Mg can be conformally immersed into R3 as a complete finite total curvature minimal surface with exactly r punctures; the set of all puncture numbers for Mg is denoted by P (M ). For example, Jorge and Meeks [JM] showed, by constructing an example g for each r, that every positive integer r is a puncture number for the Riemann surface pl.

Keywords

Minimal surface Riemann surfaces algebraic varieties crystallography curvature differential geometry homology materials science moduli space surfaces

Authors and affiliations

  • Kichoon Yang
    • 1
  1. 1.Department of MathematicsArkansas State UniversityUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-1104-3
  • Copyright Information Springer Science+Business Media B.V. 1994
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4443-3
  • Online ISBN 978-94-017-1104-3
  • Buy this book on publisher's site
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