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Surfaces with Constant Mean Curvature
Katsuei Kenmotsu, Tohoku University, Sendai, Japan

Translations of Mathematical Monographs
2003; 142 pp; softcover
Volume: 221
ISBN-10: 0-8218-3479-7
ISBN-13: 978-0-8218-3479-4
List Price: US$63
Member Price: US$50.40
Order Code: MMONO/221
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The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the three-dimensional space. A surface whose mean curvature is zero at each point is a minimal surface, and it is known that such surfaces are models for soap film. There is a rich and well-known theory of minimal surfaces. A surface whose mean curvature is constant but nonzero is obtained when we try to minimize the area of a closed surface without changing the volume it encloses. An easy example of a surface of constant mean curvature is the sphere. A nontrivial example is provided by the constant curvature torus, whose discovery in 1984 gave a powerful incentive for studying such surfaces. Later, many examples of constant mean curvature surfaces were discovered using various methods of analysis, differential geometry, and differential equations. It is now becoming clear that there is a rich theory of surfaces of constant mean curvature.

In this book, the author presents numerous examples of constant mean curvature surfaces and techniques for studying them. Many finely rendered figures illustrate the results and allow the reader to visualize and better understand these beautiful objects.

The book is suitable for advanced undergraduates, graduate students, and research mathematicians interested in analysis and differential geometry.


Advanced undergraduates, graduate students and research mathematicians interested in analysis and differential geometry.


From a review of the Japanese edition:

"The first thing one notices about this book is that it includes many beautiful pictures of surfaces, which allow the reader to move comfortably through the material. The book takes the reader from historical results through current research ... It has distinct charm ... the author's research is impressive ... has an inviting style that draws the reader to the interesting contents of the book."

-- translated from Sugaku Expositions

Table of Contents

  • Other titles in this series
  • Preliminaries from the theory of surfaces
  • Mean curvature
  • Rotational surfaces
  • Helicoidal surfaces
  • Stability
  • Tori
  • The balancing formula
  • The Gauss map
  • Intricate constant mean curvature surfaces
  • Supplement
  • Programs for the figures
  • Postscript
  • Bibliography
  • List of sources for the figures
  • Index
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