Equivariant harmonic maps associated to large group actions
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).
scientific article; zbMATH DE number 737441