Surfaces with Constant Mean Curvature
About this Title
Katsuei Kenmotsu, Tohoku University, Sendai, Japan. Translated by Professor Katsuhiro Moriya
Publication: Translations of Mathematical Monographs
Publication Year: 2003; Volume 221
ISBNs: 978-0-8218-3479-4 (print); 978-1-4704-4645-1 (online)
MathSciNet review: MR2013507
MSC: Primary 53A10; Secondary 35J60
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. 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. A trivial 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.
In this book, the author presents the numerous examples of constant mean curvature surfaces and the techniques for studying them. Many figures illustrate the presented results and allow the reader to visualize and better understand these beautiful objects.
Advanced undergraduates, graduate students and research mathematicians interested in analysis and differential geometry.
Table of Contents
- Preliminaries from the theory of surfaces
- Mean curvature
- Rotational surfaces
- Helicoidal surfaces
- The balancing formula
- The Gauss map
- Intricate constant mean curvature surfaces
- Programs for the figures