Book review

Morse Theory's object is the relation between the topological type of critical points of a map and the topological structure of the manifold on which the map is defined. The topological type of a critical point is described by the critical groups of Morse while the topological structure of the manifold is described by its Betti numbers. "This book deals with Morse theory as a way of studying multiple solutions of differential equations which arise in the calculus of variations. The theory consists of two aspects: the global one, in which existence, including the estimate of the number of solutions, is obtained by the relative homology groups of two certain level sets, and the local one, in which a sequence of groups, which we call critical groups, is attained to an isolated critical point (or orbit) to describe the local behavior of the functional".

The book is organised into five chapters and an appendix. Chapter I: Infinite dimensional Morse theory (deals with basic facts of algebraic topology and infinite dimensional manifolds, deformation theorems, critical groups and Morse type numbers, Gromoll-Meyer theory, extensions of Morse theory for general boundary conditions and to locally convex closed sets, and equivariant Morse theory); Chapter II: Critical point theory (with respect to homology groups); Chapter III: Applications to semilinear elliptic boundary value problems (some of these applications are known but the proofs given here are new and are based on the above unified framework); Chapter IV: Multiple periodic solutions of Hamiltonian systems (asymptotically linear systems, Hamiltonians with periodic nonlinearities, second order systems with singular potentials, the double pendulum equation, Arnold conjectures on symplectic fixed points and on Lagrangian intersections); Chapter V: Applications to harmonic maps and minimal surfaces (the Plateau problem); Appendix: Witten's proof of the Morse inequalities.

The book has a good list of References, an short Index of Notation and a general Index.

The volume is intented to be as self-contained as possible and to be of interest both to researchers and graduate students working in nonlinear analysis, partial differential equations, differential geometry and algebraic topology.

Written by a well known specialist in the field, the book is indeed, in addition to the remarkable up-to-date volumes: Critical Point Theory and Hamiltonian Systems, by J. Mawhin & M. Willem (Springer, 1989); Variational Methods, by M. Struwe (Springer, 1990), a valuable new contribution to the theory of the variational methods for nonlinear ordinary and partial differential equations.

We warmly recommend this excelent book to all interested in critical point theory, nonlinear differential equations and nonlinear analysis.