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

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

2008-03-08

arxiv.org/abs/0803.1249v1

J. Symp. Geom. 8 (2010), 239-265

Mathematics

Differential Geometry

Scientific paper

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.

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