Harmonic maps of bounded symmetric domains

Abstract

Harmonic maps between complete and non-compact manifolds have been studied by Schoen-Yau, Li-Tam, Aviles-Choi-Micallef, Ding-Wang and others ([S-Y] [L-T] [A-C-M] and [D-W]). For further study it might be of interest to consider problems in various concrete manifolds which are more general than hyperbolic space. Bounded symmetric domains were introduced by E. Cartan [C1] [C2] and were systematically studied by Hua, Look and Siegel [HI] [H2] [L] [Si]. Those are specific Cartan-Hadamard manifolds whose further geometrical and analytical properties should be explored. Such investigation might also imply more general results for complete and non-compact manifolds. The purpose of the present paper is to pursue this goal. By the work done by Y.C. Wong [W] a bounded symmetric domain can be viewed as a pseudo-Grassmannian manifold of all the spacelike subspaces of dimension m in pseudo-Euclidean space R~+n of index n. In Sect. 2 this interesting point of view is briefly introduced. It is well-known that the simplest bounded symmetric domain Nm(2) can be identified with a product of two hyperbolic planes [He]. In Sect. 3 we study harmonic maps into Nn,(2) via harmonic maps into hyperbolic space and obtain an interesting image shrinking property. In the striking paper [H-O-S] the authors classified all complete constant mean curvature surfaces in R 3 and ~4 with certain restricted Gauss image. One of the analogous properties in ambient Minkowski space was already proved by a different approach in the author's previous paper [XI] (also see [A]). By Ruh-Vilms theorem [R-V] we can investigate submanifolds with parallel mean curvature in Euclidean space via its harmonic Gauss maps. In the present