Cohomogeneity One Einstein-sasaki 5-manifolds

We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one. From a Riemannian point of view, an Einstein-Sasaki manifold is a Riemannian manifold (M, g) such that the conical metric on M × R is Kähler and Ricci-flat. In particular, this implies that (M, g) is odd-dimensional, contact and Einstein with positive scalar curvature. The Einstein-Sasaki manifolds that are simplest to describe are the regular ones, which arise as circle bundles over Kähler-Einstein manifolds. In five dimensions, there is a classification of regular Einstein-Sasaki manifolds [11], in which precisely two homogeneous examples appear, namely the sphere S and the Stiefel manifold (1) V2,4 = SO(4)/SO(2) ∼= S × S . In fact these examples are unique (up to finite cover), as homogeneous contact manifolds are necessarily regular [3]. Among regular Einstein-Sasaki 5-manifolds, these two are the only ones for which the metric is known explicitly: indeed, the sphere is equipped with the standard metric, and the metric on V2,4 has been described in [20]. Notice however that both S and S × S carry other, nonregular Einstein-Sasaki metrics [4, 5]. Only recently other explicit examples of Einstein-Sasaki manifolds have been found, as in [12] the authors constructed an infinite family Y p,q of Einstein-Sasaki metrics on S × S (see also [9] for a generalization). These metrics are non-regular, and thus they are not included in the above-mentioned classification. The isometry group of each Y p,q acts with cohomogeneity one, meaning that generic orbits are hypersurfaces. In this paper we give an alternative construction of the Y , based on the language of cohomogeneity one manifolds. In fact we prove that, up to finite cover, they are the only Einstein-Sasaki 5-manifolds on which the group of isometries acts with cohomogeneity one. In particular, this result settles a question raised in [13], concerning the family of links L(2, 2, 2, k), k > 0 defined by the polynomial z 1 + z 2 2 + z 2 3 + z k 4 . The homogeneous metrics mentioned earlier provide examples of Einstein-Sasaki metrics on L(2, 2, 2, k), k = 1, 2 such that the integral lines of the characteristic vector field are the orbits of the natural action of U(1) with weights (k, k, k, 2). The authors of [13] show that for k > 3 no such metric exists. Since L(2, 2, 2, 3) 2000 Mathematics Subject Classification. Primary 53C25; Secondary 53C30, 57S15. 1