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September 2010 Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties
Yanir A. Rubinstein, Steve Zelditch
J. Symplectic Geom. 8(3): 239-265 (September 2010).


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.


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Yanir A. Rubinstein. Steve Zelditch. "Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties." J. Symplectic Geom. 8 (3) 239 - 265, September 2010.


Published: September 2010
First available in Project Euclid: 7 September 2010

zbMATH: 1205.53067
MathSciNet: MR2684507

Rights: Copyright © 2010 International Press of Boston


Vol.8 • No. 3 • September 2010
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