Bergman Approximations of Harmonic Maps into the Space of Kähler Metrics on Toric Varieties

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 C topology by harmonic maps into the spaces of Bergman metrics. In particular, WZW maps, or equivalently solutions of a homogeneous Monge-Ampère equation on the product of the manifold with a Riemann surface with S 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.