We generalize the results of Song–Zelditch on geodesics in spaces
of Kähler metrics on toric varieties to harmonic maps of any compact
Riemannian manifold with boundary into the space of Kähler metrics
on a toric variety. We show that the harmonic map equation can
always be solved and that such maps may be approximated in the
$C2$ topology by harmonic maps into the spaces of Bergman metrics. In
particular, Wess–Zumino–Witten (WZW) maps, or equivalently solutions
of a homogeneous Monge–Ampère equation on the product of the
manifold with a Riemann surface with $S1$ boundary admit such approximations.
We also show that the Eells–Sampson flow on the space of
Kähler potentials is transformed to the usual heat flow on the space
of symplectic potentials under the Legendre transform, and hence it
exists for all time and converges.