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Elements of the geometry and topology of minimal surfaces in three-dimensional space
Edited by: A T Fomenko and Avgustin Tuzhilin

Translations of Mathematical Monographs
1991; 142 pp; softcover
Volume: 93
Reprint/Revision History:
reprinted 2005
ISBN-10: 0-8218-3791-5
ISBN-13: 978-0-8218-3791-7
List Price: US$59
Member Price: US$47.20
Order Code: MMONO/93.S
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This book grew out of lectures presented to students of mathematics, physics, and mechanics by A. T. Fomenko at Moscow University, under the auspices of the Moscow Mathematical Society. The book describes modern and visual aspects of the theory of minimal, two-dimensional surfaces in three-dimensional space. The main topics covered are: topological properties of minimal surfaces, stable and unstable minimal films, classical examples, the Morse-Smale index of minimal two-surfaces in Euclidean space, and minimal films in Lobachevskian space. Requiring only a standard first-year calculus and elementary notions of geometry, this book brings the reader rapidly into this fascinating branch of modern geometry.


"Plenty of motivational examples. The prerequisites are kept to a minimum - a large portion of the book should be accessible to undergraduate mathematics students. Clear and elementary proofs coupled with illuminating illustrations make the book pleasant reading."

-- Mathematical Reviews

"Serves as a beautiful introduction to the subject of minimal surfaces."

-- Zentralblatt MATH

Table of Contents

  • Physical prerequisites: Interfaces between two media
  • The principle of economy in nature
  • Classical minimal surfaces in \(R^3\): Catenoids
  • The helicoid
  • The minimal surface equation, Bernstein's problem, The Scherk surface
  • Periodic minimal surfaces
  • Complete minimal surfaces
  • General properties of minimal surfaces in \(R^3\): Isothermal coordinates
  • Harmonicity and conformality
  • The Gaussian mapping and the Weierstrass representation
  • The global Weierstrass representation
  • Total curvature and complete minimal surfaces
  • The geometry of complete minimal surfaces of finite total curvature
  • Indices of two-dimensional minimal surfaces in \(R^3\)
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