Hermitian Curvature and Plurisubharmonicity of Energy on Teichmüller Space

Let M be a closed Riemann surface, N a Riemannian manifold of Hermitian non-positive curvature, f : M → N a continuous map, and E the function on the Teichmüller space of M that assigns to a complex structure on M the energy of the harmonic map homotopic to f. We show that E is a plurisubharmonic function on the Teichmüller space of M. If N has strictly negative Hermitian curvature, we characterize the directions in which the complex Hessian of E vanishes.