Geometry of harmonic maps
The theory of harmonic maps is a very much alive research domain in todays mathematics. It grew out from several essential notions in differential geometry, such as: geodesics, minimal surfaces, harmonic functions, holomorphic maps between Kähler manifolds, etc. The theory of harmonic maps is closely related to the differential geometry of complex manifolds, the nonlinear field theory in theoretical physics, the theory of stochastic processes, the theory of liquid crystals in materials science, etc. The notion of harmonic map has been introduced in 1964 by J. Eells and J. H. Sampson by using a variational principle for the energy functional. During the past thirty years the theory has been developed extensively. Now, there are at least eight monographs dealing with this argument and two survey papers by J. Eells and L. Lemaire. The present book provides an excellent introduction to the theory of harmonic maps, being useful to research workers and graduate students in differential geometry, geometric analysis, differential equations and theoretical physics. The author presents the geometric aspects of harmonic maps and establishes the inter-relationship with other mathematical and physical topics. The author's own contributions to some topics presented in the book are described. The first chapter contains some introductory material. The author presents several equivalent definitions of harmonic maps and some interesting examples, obtaining various important properties and formulas. A Bochner type formula is obtained by using the Weitzenböck formula. Some basic properties of harmonic maps such as the maximum principle, unique continuation theorems, second variational formula and the stability of the harmonic maps are discussed. Next, in the second chapter, he obtains the conservation laws for harmonic maps and presents some applications to the monotonicity formula and Liouville type theorems by using the stress-energy tensor as well as further generalizations. In the third chapter there are given some uniqueness results for submanifolds with parallel mean curvature through the property of their Gauss maps to be harmonic. After obtaining the generalized maximum principle and some estimates for the image diameter, the author studies the properties of the Gauss images of a space-like hypersurface in a Minkowski space and of a space-like submanifold in a pseudo-Euclidean space. Then, some aspects related to harmonic maps in complex geometry are presented in the fourth chapter. Some applications of harmonic maps to the Frankel conjecture and to strong rigidity are given. The existence results for harmonic maps are discussed in the fifth chapter. The author is interested in finding the geometric conditions under which he can apply the partial regularity theorems of Schoen-Uhlenbeck and Giaquinta-Giusti. In this way, it is obtained a generalization of the Eells-Sampson existence theorem. In the last chapter the author makes a systematic study of the equivariant harmonic maps. After a short presentation of the Riemannian submersions and the equivariant harmonic maps, the author obtains the reduction theorems and equivariant variational formulas, considering the cases where the reduced harmonicity equations are a single ordinary differential equation, a system of ordinary differential equations or a scalar partial differential equation. Finally he solves several concrete problems, showing that the method is useful for finding nonminimal critical points of some geometric variational problems. In conclusion one can say that this is a good and useful book worth of finding its place in any mathematician's bookcase who is interested in the theory of harmonic maps.
scientific article; zbMATH DE number 877981