Journal of Symplectic Geometry

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

Yanir A. Rubinstein and Steve Zelditch

Full-text: Open access

Abstract

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.

Article information

Source
J. Symplectic Geom., Volume 8, Number 3 (2010), 239-265.

Dates
First available in Project Euclid: 7 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.jsg/1283865583

Mathematical Reviews number (MathSciNet)
MR2684507

Zentralblatt MATH identifier
1205.53067

Citation

Rubinstein, Yanir A.; Zelditch, Steve. Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties. J. Symplectic Geom. 8 (2010), no. 3, 239--265. https://projecteuclid.org/euclid.jsg/1283865583


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