-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 | References | Similar Articles | Additional Information

Abstract: In this note, we prove that on polarized toric manifolds the relative -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 -energy is proper in the space of -invariant Kähler potentials in the case of toric surfaces which admit the extremal metrics.

**[Ab]**Miguel Abreu,*Kähler geometry of toric varieties and extremal metrics*, Internat. J. Math.**9**(1998), no. 6, 641–651. MR**1644291**, 10.1142/S0129167X98000282**[D1]**S. K. Donaldson,*Scalar curvature and projective embeddings. I*, J. Differential Geom.**59**(2001), no. 3, 479–522. MR**1916953****[D2]**S. K. Donaldson,*Scalar curvature and stability of toric varieties*, J. Differential Geom.**62**(2002), no. 2, 289–349. MR**1988506****[Gu]**Daniel Guan,*On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles*, Math. Res. Lett.**6**(1999), no. 5-6, 547–555. MR**1739213**, 10.4310/MRL.1999.v6.n5.a7**[M1]**Toshiki Mabuchi,*An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds. I*, Invent. Math.**159**(2005), no. 2, 225–243. MR**2116275**, 10.1007/s00222-004-0387-y**[M2]**Mabuchi, T., An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II, preprint, 2004. arxiv:math/0410239**[Sz]**Gábor Székelyhidi,*Extremal metrics and 𝐾-stability*, Bull. Lond. Math. Soc.**39**(2007), no. 1, 76–84. MR**2303522**, 10.1112/blms/bdl015**[Ti]**Gang Tian,*Kähler-Einstein metrics with positive scalar curvature*, Invent. Math.**130**(1997), no. 1, 1–37. MR**1471884**, 10.1007/s002220050176**[Ya]**Shing-Tung Yau,*Open problems in geometry*, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1–28. MR**1216573****[ZZ1]**Zhou, B., and Zhu, X.H., Relative K-stability and modified K-energy on toric manifolds, to appear in Advances in Math.**[ZZ2]**Zhou, B., and Zhu, X.H., Minimizing weak solutions for calabi's extremal metrics on toric manifolds, Calculus in Variations and PDE, 32 (2008), 191-217.

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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

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