Uniqueness and examples of compact toric Sasaki-Einstein metrics

Abstract

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 $S^5♯k(S^2\times S^3)$ for each positive integer $k$.

Statements of the results are modified

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• Ricci-flat Kähler metrics on crepant resolutions of Kähler cones
• Examples of asymptotically conical Ricci-flat Kähler manifolds
• On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$
• Weighted K-stability of polarized varieties and extremality of Sasaki manifolds
• Special Killing forms on toric Sasaki-Einstein manifolds
• Self-similar solutions to the mean curvature flows on Riemannian cone manifolds and special Lagrangians on toric Calabi-Yau cones
• Coupled Sasaki-Ricci solitons
• Extremal Sasakian Metrics on S^3-bundles over S^2
• Hilbert series and obstructions to asymptotic semistability
• Uniqueness of Constant Scalar Curvature Sasakian Metrics
• Special Lagrangians and Lagrangian self-similar solutions in cones over toric Sasaki manifolds
• On irregular Sasaki-Einstein metrics in dimension 5
• Momentum construction on Ricci-flat Kähler cones
• Some topics on Ricci solitons and self-similar solutions to mean curvature flow
• Regularity of the geodesic equation in the space of Sasakian metrics
• Toric data and Killing forms on homogeneous Sasaki-Einstein manifold $T^{1,1}$
• $G$-Sasaki manifolds and K-energy
• New examples of Sasaki-Einstein manifolds
• Special Legendrian submanifolds in toric Sasaki-Einstein manifolds