  We prove that the solution of a Wess-Zumino-Witten type equation from a
domain $D$ in $\mathbb{C}^m$ to the space of K\"ahler potentials can be
approximated uniformly by Hermitian-Yang-Mills metrics on certain vector
bundles. The key is a new version of Berndtsson's theorem on the positivity of
direct image bundles.
