This book provides a wide view of the calculus of variations as it plays an essential role in various areas of mathematics and science. Containing many examples, open problems, and exercises with complete solutions, the book would be suitable as a text for graduate courses in differential geometry, partial differential equations, and variational methods. The first part of the book is devoted to explaining the notion of (infinite-dimensional) manifolds and contains many examples. An introduction to Morse theory of Banach manifolds is provided, along with a proof of the existence of minimizing functions under the Palais-Smale condition. The second part, which may be read independently of the first, presents the theory of harmonic maps, with a careful calculation of the first and second variations of the energy. Several applications of the second variation and classification theories of harmonic maps are given.

Readership

Graduate students and research mathematicians. Containing many examples, open problems, and exercises with complete solutions, the book is suitable as a text for graduate courses in differential geometry, partial differential equations, and variational methods.

Reviews

"This book is organized in an elegant manner. The background and motives of the subjects concerned are clearly explained. A fundamental knowledge of differential geometry is also included and the exercises at the end of each chapter are useful for the readers to deepen their understandings. It is a good book for one who wants to know and to study the recent developments in the theories of harmonic maps and the variational methods."

-- Zentralblatt MATH

"This is probably the first textbook on the theory of harmonic maps ... It is ... an extremely welcome addition to the literature, both as an introductory text and as a reference for those details which are often taken for granted in research articles."

-- Mathematical Reviews

Table of Contents

• Calculus of variations
• Manifolds
• Morse theory
• Harmonic mappings
• The second variation formula and stability
• Existence, construction, and classification of harmonic maps
• Solutions to exercises
• References
• Subject index