Harmonic maps from Kähler manifolds (Q1292806)

This paper may be viewed as the continuation of the previous paper of the author [Math. Ann. 303, No. 3, 417-433 (1995; Zbl 0842.58015)]. In that paper, by applying a useful integral identity for any smooth map and pointwise estimates for the Hessian of the distance function in a Cartan-Hadamard manifold, the author proved a Liouville type theorem of a harmonic map from a Cartan-Hadamard manifold to any Riemannian manifold with finite energy under certain curvature conditions. In this paper, the author improves the dimension limitation on the results of that paper. Precisely, the following theorem is proved: A harmonic map of finite energy from a classical bounded symmetric domain (except \({\mathcal R}_{\text{IV}}(2)= \mathbb{H}^2\times \mathbb{H}^2\)) to any Riemannian manifold has to be constant. The proof of this theorem is based on the following proposition: Let \(M\) be a simply connected complete Kähler manifold of real dimension \(2n\geq 4\) with nonpositive sectional curvature. Assume that the geodesic spheres centered at a fixed point satisfy the following conditions: (1) \(J{\partial\over\partial r}\) is of a principal direction with respect to the principal curvature \(-\widetilde\lambda_n\) for the geodesic sphere \(S_R(x_0)\) in \(M\); (2) If \(e_\alpha\), for each \(\alpha\), is of a principal direction with respect to the principal curvature \(-\lambda_\alpha\), so is the \(Je_\alpha\) with respect to the principal curvature \(-\widetilde\lambda_\alpha\). Then any harmonic map from \(M\) to any Riemannian manifold with finite energy has to be constant.