Uniqueness of Sasaki-Einstein metrics

Abstract

In this paper, we shall prove the uniqueness of Sasaki-Einstein metrics on compact Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure. This generalizes the result of Cho, Futaki and Ono [5] for compact toric Sasaki manifolds. Introduction.The aim of this paper is to show the uniqueness of Sasaki-Einstein metrics up to the action of the identity component of the automorphism group for the transverse holomorphic structure. A Sasaki manifold is a Riemannian manifold (S, g) whose cone metric ḡ = dr 2 + r 2 g on C(S) = S × R + is Kählerian. Sasakian geometry sits naturally in two aspects of Kähler geometry, since for one thing, (S, g) is the base of the cone manifold (C(S), ḡ ) which is Kählerian, and for another thing any Sasaki manifold is contact, and the one dimensional foliation associated to the characteristic Reeb vector field admits a transverse K ähler structure. A Sasaki-Einstein manifold then admits a one dimensional Reeb foliation with a transverse Kähler-Einstein metric, which is studied from viewpoints of geometry and mathematical physics. Boyer, Galicki, Kollár and Thomas obtained Sasaki-Einstein metrics on a family of the links of hypersurfaces of Brieskorn-Pham type, which include exotic spheres [3, 4]. Gauntlett, Martelli, Sparks and Waldram discovered that there exist irregular toric Sasaki-Einstein manifolds which are not obtained as total spaces of line orbibundles on Kähler-Einstein orbifolds [8, 9]. These toric examples are much explored by Futaki, Ono and Wang [7], who showed that, for any compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasakian structure on which a Sasaki-Einstein metric exists. Furthermore, Cho, Futaki and Ono proved in [5] the uniqueness of Sasaki-Einstein metrics on compact toric Sasaki manifolds up to the action of the identity component of the automorphism group for the transverse holomorphic structure by showing that the argument of Guan [10] is valid also for the space of Kähler potentials for the transverse Kähler structure. In the present paper, we shall prove such uniqueness without toric assumption: THEOREM A. Let (S, g) be a compact Sasaki manifold with a Sasakian structure S = {g, ξ, η, Φ}. Assume that the set E of all Sasaki-Einstein metrics compatible with g is non-empty. Then the identity component of the automorphism group for the transverse holomorphic structure acts transitively on E .

Quasi-asymptotically conical Calabi–Yau manifolds

“…]. If we fix the complex manifold (C, J 0 ) and the scaling vector field r∂ r , then the Calabi-Yau cone structure (g 0 , Ω 0 ) is unique up to scaling and up to the action of Aut T 0 (C) [78,79]. The links of Calabi-Yau cones are called Sasaki-Einstein manifolds.…”

The Sasaki join and admissible Kähler constructions

“…A special case of particular interest are the Y p,q structures on S 2 × S 3 which are discussed further in Example 7.6 below. The uniqueness statement in the theorem follows from [NS12] which proves transverse uniqueness, up to transverse holomorphic transformations, then using [MSY08] which proves uniqueness in the Sasaki cone. 6.3.…”

On the convergence of the Sasaki J-flow

“…(in our notation the basic Laplacian has the opposite sign of [17] one). Thus, for any ψ ∈ C ∞ B (M, R) we have:…”