  We show that if $\{M_t\}_{t\in \Delta}$ is a polarized family of compact
K\"ahler manifolds over the open unit disk $\Delta$, if $N$ is a Riemannian
manifold of nonpositive complexified sectional curvature, and if
$\{\phi_t:M_t\to N\}_{t\in \Delta}$ is a smooth family of pluriharmonic maps,
then the Dirichlet energy $E(\phi_t)$ is a subharmonic function of
$t\in\Delta$.
