On proper harmonic maps between strictly pseudoconvex domains with Kähler metrics of Bergman type

where (h) is the inverse of the matrix (hij), ∆M = ∑ i,j h ∂ij and Γ s tγ denote the Christoffel symbols of the Hermitian metric g on N . It follows from (1.1) that if u is holomorphic, then u must be harmonic. Thus, it is natural to ask under what circumstances a harmonic map is holomorphic or antiholomorphic. Under the assumption that both M and N are compact, Siu [31] demonstrated that if the curvature tensor of N is strongly negative and the rank of du is greater than or equal to four at a point of M , then a harmonic map u must be holomorphic or antiholomorphic. The proof follows from Siu’s Bochner type identity together with the compactness assumption on M . If M is a complete noncompact manifold of strongly negative curvature with infinite volume, the previous Bochner type identity technique fails and