A Wess–Zumino–Witten type equation in the space of Kähler potentials in terms of Hermitian–Yang–Mills metrics

We prove that the solution of a Wess-Zumino-Witten type equation from a domain D in Cm to the space of Kähler 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.