Uniqueness and examples of compact toric Sasaki-Einstein metrics

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

2007-01-04

arxiv.org/abs/math/0701122v3

Commun.Math.Phys.277:439-458,2008

Mathematics

Differential Geometry

Statements of the results are modified

Scientific paper

10.1007/s00220-007-0374-4

In [11] it was proved that, given a compact toric Sasaki manifold of 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 $S^5 \sharp k(S^2 \times S^3)$ for each positive integer $k$.

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