|Title||Complete and Compact Minimal Surfaces|
|Publisher||Kluwer Academic Publishers|
|Year of publication||1989|
|Reviewed by||VALERII V. TROFIMOV|
The book under review is devoted to minimal surfaces. Let <formula> be the surface with equation <formula> into Euclidean space <formula>. Then, one is minimal if relation <formula> is valid. There is the generalization of this condition to the arbitrary Riemannian manifold N^n and its submanifolds <formula>. This area constitutes one of the important branch of the modern geometry, mechanics and mathematical physics. At the present time, we have many books elucidated this field, see, for example, [1, 2]. The Yang's book is a very good supplement to literature in existence.
The following topics are presented in the book under review. The first chapter, Complete Minimal Surfaces in R^n, provides known results on minimal surfaces in R^n. The second chapter, Compact Minimal Surfaces in S^n, discusses the structure of the compact minimal surfaces in sphere S^n with moving frames stand point. The third chapter, Holomorphic Curves and Minimal Surfaces in CP^n examines the holomorphic curves in complex projective space CP^n and related minimal surfaces. The fourth chapter, Holomorphic Curves and Minimal Surfaces in the Quadric, supplies a generalization to complex quadrics and flag manifolds of the works of Calabi, Bryant, Eells-Wood, Chern-Wolfson (presented in the 2 th and 3 th chapters) from uniform stand point. The fifth chapter, The Twistor Method, gives introduction to twistor method and its application to minimal surface theory.
The level is post-graduated. The specialists in the minimal surfaces theory can find in this book very many interesting topics as well. On the other hand, the geometric methods presented in the book are attractive to the general readers with interest in modern geometry.
All in all, this book has to find the wide range of buyers. It is an extremely valuable contribution to the literature on minimal surfaces. Purchasing of Complete and Compact Minimal Surfaces is a good investment for researchers, students, and libraries alike.
1. Fomenko, A.T.: Variational Problems in Topology. The Geometry of Length, area and Volume, Gordon and Breach Science Publishers, New York at all, 1990.
2. Fomenko, A.T., Tuzhilin, A.A., Elements of the Geometry and Topology of minimal Surfaces in Three-Dimensioal Space, American Mathematical Society, Providence, Rhode Island, 1991