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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ K$-stability on toric manifolds


Authors: Bin Zhou and Xiaohua Zhu
Journal: Proc. Amer. Math. Soc. 136 (2008), 3301-3307
MSC (2000): Primary 53C25; Secondary 32J15, 53C55, 58E11
Published electronically: April 29, 2008
MathSciNet review: 2407096
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Abstract: In this note, we prove that on polarized toric manifolds the relative $ K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and we also show that the modified $ K$-energy is proper in the space of $ G_0$-invariant Kähler potentials in the case of toric surfaces which admit the extremal metrics.


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Additional Information

Bin Zhou
Affiliation: Department of Mathematics, Peking University, Beijing, 100871, People’s Republic of China

Xiaohua Zhu
Affiliation: Department of Mathematics, Peking University, Beijing, 100871, People’s Republic of China
Email: xhzhu@math.pku.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09485-9
PII: S 0002-9939(08)09485-9
Keywords: $K$-stability, toric manifolds, extremal metrics
Received by editor(s): July 17, 2007
Published electronically: April 29, 2008
Additional Notes: The second author was partially supported by NSF10425102 in China.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2008 American Mathematical Society