**Translations of Mathematical Monographs**

2003;
142 pp;
Softcover

MSC: Primary 53;

**Print ISBN: 978-0-8218-3479-4
Product Code: MMONO/221**

List Price: $72.00

AMS Member Price: $57.60

MAA Member Price: $64.80

**Electronic ISBN: 978-1-4704-4645-1
Product Code: MMONO/221.E**

List Price: $67.00

AMS Member Price: $53.60

MAA Member Price: $60.30

#### Supplemental Materials

# Surfaces with Constant Mean Curvature

Share this page
*Katsuei Kenmotsu*

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.

#### Readership

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

#### Reviews & Endorsements

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

# Table of Contents

## Surfaces with Constant Mean Curvature

- Cover Cover11
- Title page i2
- Frontispiece iii4
- Foreword v6
- Contents ix10
- Preliminaries from the theory of surfaces 112
- Mean curvature 2536
- Rotational surfaces 3950
- Helicoidal surfaces 5364
- Stability 6374
- Tori 7586
- The balancing formula 8394
- The Gauss map 97108
- Intricate constant mean curvature surfaces 109120
- Supplement 121132
- Programs for the figures 127138
- Postscript 133144
- Bibliography 137148
- List of sources for the figures 139150
- Index 141152
- Back Cover Back Cover1154