Hermitian harmonic maps from complete Hermitian manifolds to complete Riemannian manifolds

Abstract

In this paper we study a nonlinear elliptic system of equations imposed on a map from a complete Hermitian (non-Kahler) manifold to a Riemannian manifold. This system is more appropriate to Hermitian geometry than the harmonic map system since it is compatible with the holomorphic structure of the domain manifold in the sense that holomorphic maps are Hermitian harmonic maps. It was first studied by Jost and Yau in [J-Y], and was applied to study the rigidity of compact Hermitian manifolds. We extend their existence and uniqueness results to the case where both domain and target manifolds are complete. Hopefully the results will be useful to study corresponding rigidity of complete Hermitian manifolds. Let M be a complex manifold with Hermitian metric (hαβ), and let N be a Riemannian manifold with metric (gij) and Christoffel symbols Γ i jk. A Hermitian harmonic map u : M → N satisfies the following elliptic system