### Abstract

In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on S^{5} # k(S^{2} × S^{3}) for each positive integer k.

Original language | English |
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Pages (from-to) | 439-458 |

Number of pages | 20 |

Journal | Communications in Mathematical Physics |

Volume | 277 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2008 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*277*(2), 439-458. https://doi.org/10.1007/s00220-007-0374-4

**Uniqueness and examples of compact toric Sasaki-Einstein metrics.** / Cho, Koji; Futaki, Akito; Ono, Hajime.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 277, no. 2, pp. 439-458. https://doi.org/10.1007/s00220-007-0374-4

}

TY - JOUR

T1 - Uniqueness and examples of compact toric Sasaki-Einstein metrics

AU - Cho, Koji

AU - Futaki, Akito

AU - Ono, Hajime

PY - 2008/1/1

Y1 - 2008/1/1

N2 - In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on S5 # k(S2 × S3) for each positive integer k.

AB - In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on S5 # k(S2 × S3) for each positive integer k.

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UR - http://www.scopus.com/inward/citedby.url?scp=36448984390&partnerID=8YFLogxK

U2 - 10.1007/s00220-007-0374-4

DO - 10.1007/s00220-007-0374-4

M3 - Article

AN - SCOPUS:36448984390

VL - 277

SP - 439

EP - 458

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -