  We generalize the results of Song-Zelditch on geodesics in spaces of Kahler
metrics on toric varieties to harmonic maps of any compact Riemannian manifold
with boundary into the space of Kahler 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 C^2 topology by harmonic maps into the spaces of Bergman
metrics. In particular, WZW maps, or equivalently solutions of a homogeneous
Monge-Ampere equation on the product of the manifold with a Riemann surface
with S^1 boundary admit such approximations. We also show that the
Eells-Sampson flow on the space of Kahler potentials is transformed to the
usual heat flow on the space of symplectic potentials under the Legendre
transform.
