Geometry of harmonic maps
type: Item
instance of: scholarly article
title: Geometry of harmonic maps
zbMATH Open document ID: 0848.58014
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 1996-05-13T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 58E30
zbMATH DE Number: 877981
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: variational methods
zbMATH Keywords: equivariant maps
zbMATH Keywords: harmonic maps
zbMATH Keywords: Kähler manifolds
author: Yuanlong Xin
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 877981