Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties

TL;DR

This paper extends the approximation of harmonic maps into Kähler metric spaces on toric varieties using Bergman metrics, demonstrating solutions exist and can be approximated, with applications to WZW maps and heat flow transformations.

Contribution

It generalizes previous results to harmonic maps from any compact Riemannian manifold with boundary into Kähler metrics on toric varieties, including new approximation techniques.

Findings

Harmonic maps can be approximated by Bergman metric maps in C^2 topology.

Solutions exist for harmonic map equations into Kähler metric spaces.

Eells-Sampson flow corresponds to heat flow under Legendre transform.

Abstract

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.