  Let $M$ be a closed Riemann surface, $N$ a Riemannian manifold of Hermitian
non-positive curvature, $f:M\to N$ a continuous map, and $E$ the function on
the Teichm\"uller 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 Teichm\"uller space of $M$. If $N$ has
strictly negative Hermitian curvature, we characterize the directions in which
the complex Hessian of $E$ vanishes.
