Infinite dimensional Morse theory and its applications. (English) Zbl 0609.58001

Chang, Kung-Ching

Séminaire de Mathématiques Supérieures. Séminaire Scientifique OTAN (NATO Advanced Study Institute), 97. Département de Mathématiques et de Statistique, Université de Montréal. Montréal (Québec), Canada: Les Presses de l’Université de Montréal. 154 p. $ 19.00 (1985).

From the introduction: ”The aim of these lectures is to introduce the Morse theory systematically, and to emphasize its applications in studying multiple solution problems in nonlinear analysis. Sections 1 and 2 present the basic theory on Hilbert Riemannian manifolds. The two equivalent definitions of critical groups, the Morse lemma and its generalization - the splitting theorem, the Morse inequalities and a shifting theorem on isolated degenerate critical points, are studied. Section 3 extends the classical Morse theory to Banach-Finsler manifolds. The nondegeneracy is defined, and the Morse inequalities are established. The interplay between the Leray-Schauder index and the critical groups is studied in Section 4 as a local version of the Poincaré-Hopf theorem. In the same section, a formula connecting the Leray-Schauder degree with the relative Euler characteristic, and a minimax theorem generalizing the Lyusternik-Shnirel’man category theorem via the cap product of a cohomology class with relative homology classes are obtained. In Section 5 a three-critical-point theorem with applications is discussed. Section 6 includes a homotopy invariance theorem of the critical groups, with an application to a bifurcation theorem due to Krasnosel’skij and Rabinowitz. Section 7 studies the homological characterization of the mountain pass point. According to this characterization a more general form of the mountain pass theorem is given, which covers new applications.
Sections 8-10 deal with applications to elliptic BVP and periodic solutions of Hamiltonian systems. The reader will find that there are many different and very interesting results in these sections. Some of them have been published in the literature, but the proofs given here are new and are presented in a unified way. Some of these results are published here for the first time. Section 8 contains some abstract theorems in the functional analytic framework. They will be used in Section 9 and 10. In Section 8, there is a finite dimensional reduction theory, which is a variant of the so-called saddle point reduction due to Amann. The reader will find that the procedure of reduction has been considerably simplified. In Section 10, there is a simple proof of Arnold’s conjecture on the number of fixed points of symplectic maps.”
These are excellent notes, and are highly recommended.

Reviewer: J.Eells

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E07	Variational problems in abstract bifurcation theory in infinite-dimensional spaces
35J60	Nonlinear elliptic equations
47J05	Equations involving nonlinear operators (general)
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

Morse theory; multiple solution problems in nonlinear analysis; Hilbert Riemannian manifolds; critical groups; Morse inequalities; Leray-Schauder index; minimax theorem; Lyusternik-Shnirel’man category; homotopy invariance theorem; bifurcation theorem; homological characterization of the mountain pass point; mountain pass theorem; applications to elliptic BVP; periodic solutions of Hamiltonian systems; finite dimensional reduction theory; saddle point reduction