Equivariant harmonic maps associated to large group actions
type: Item
instance of: scholarly article
title: Equivariant harmonic maps associated to large group actions
zbMATH Open document ID: 0832.53033
author: Hajime Urakawa
author: Keisuke Ueno
published in: Tokyo Journal of Mathematics
publication date: 1996-03-12T00:00:00Z
review text: There are very few explicit constructions of harmonic maps between non- compact Riemannian manifolds, except for works by \textit{P. Baird} [Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Pitman (1983; Zbl 0515.58010)] and \textit{A. Kasue} and \textit{T. Washio} [Osaka J. Math. 27, No. 4, 899-928 (1990; Zbl 0717.58017)]. Baird reduced the harmonic map equation to an ordinary differential equation defined on \((0,\infty)\), and investigated the behavior of a solution at the origin. Kasue and Washio studied the existence of global solutions for some warped product. Yet they did not prove the existence of equivariant harmonic maps between hyperbolic spaces. The purpose of the paper under review is to extend the results of \textit{H. Urakawa} [Mich. Math. J. 40, No. 1, 27-51 (1993; Zbl 0787.58010)] to non-compact cohomogeneity one Riemannian manifolds, and to construct harmonic maps of several non-compact Riemannian manifolds including the standard Euclidean space, the hyperbolic space, the standard complex Euclidean space, and the complex hyperbolic space. These manifolds have large Lie group actions which are cohomogeneity one Riemannian manifolds. In the present paper, making use of these group actions, the authors give a set-up for reducing the harmonic map equation to an ordinary differential equation, and construct new harmonic maps (see Theorem 6.5 and Corollary 6.6).
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58E40
zbMATH DE Number: 737441
zbMATH Keywords: harmonic map equation
zbMATH Keywords: cohomogeneity one Riemannian manifolds
zbMATH Keywords: new harmonic maps
reviewed by: Radu Iordănescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270127963
OpenAlex ID: W2067697055
DOI: 10.3836/TJM/1270127963
description: scientific article; zbMATH DE number 737441