Surfaces with Constant Mean Curvature

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.

Ever since the nontrivial example of the constant curvature torus was discovered in 1984, many other examples of constant but nonzero mean curvature surfaces have been discovered using analysis, differential geometry, and differential equations. This book presents numerous examples of these surfaces and techniques for studying them. Chapters include defining smooth surfaces, the mathematical and physical meaning of the mean curvature, study of constant mean curvature surfaces via Gauss maps, and existence theorems for surfaces with or without boundaries. Knowledge of calculus (at least through Green's theorem), linear algebra, and elementary differential geometry is recommended. The book is reprinted with minor revisions from Kenmotsu's 1942 monograph, and partly based on his lectures at Tohoku U., Kanazawa U., and Federal U. of Ceará. Annotation ©2004 Book News, Inc., Portland, OR