Uniqueness and Examples of Compact Toric Sasaki-Einstein Metrics
Uniqueness and Examples of Compact Toric Sasaki-Einstein Metrics
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DOI：
10.1007/s00220-007-0374-4
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发表时间：
2007-01
影响因子：
2.4
通讯作者：
K. Cho;A. Futaki;Hajime Ono
中科院分区：
文献类型：
--
作者：
K. Cho;A. Futaki;Hajime Ono
文[11]证明了，给定一个紧致环面Sasaki流形，其基本第一Chern类为正，而接触丛的第一Chern类为平凡，则可以找到一个变形Sasaki结构，在该结构上存在Sasaki-Einstein度量。本文首先证明了紧复曲面Sasaki流形上这类Einstein度量的唯一性，其次指出[11]的结果暗示了由任意高度的复曲面图得到的所有紧Sasaki流形上相容Einstein度量的存在性，或等价地在所有锥具有平坦标准丛的紧环面Sasaki流形上。我们进一步证明了对于每一个正整数，存在一个无限族的不等价的复曲面Sasaki-Einstein度量。
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 onfor each positive integerk.