arXiv:1103.0817v1 [math.DG] 4 Mar 2011 EXAMPLES OF EINSTEIN MANIFOLDS IN ODD DIMENSIONS DEZHONG CHEN Abstract. We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano K¨ ahler-Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. Also we construct positive Einstein metrics on certain 3-sphere bundles over a Fano K¨ ahler-Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base is the complex projective plane. 1. Introduction An Einstein manifold is a (pseudo-)Riemannian manifold whose Ricci tensor is proportional to the metric tensor [4], i.e., Ric(g)= λg. (1.1) From the analytic point of view, the Einstein equation (1.1) is a complicated non-linear system of partial differential equations, and it is hard to prove the existence of Einstein metrics on an arbitrary manifold. For example, it is still unknown whether every closed manifold of dimension greater than 4 carries at least one Einstein metric [4, 0.21]. Thus people turn to studying Einstein manifolds with large isometry group, e.g., when the isometry group acts on the manifold transitively, and hence (1.1) reduces to a system of algebraic equations, or when the isometry group acts on the manifold with principal orbits of codimension one, and hence (1.1) reduces to a system of ordinary differential equations. In both cases, it becomes much more manageable to establish some existence results for Einstein metrics (see the surveys [4, Chap 7] and [30, §§2,4]). Recent progress in this direction includes the variational approach to study homogenous Einstein metrics by several authors (see [21] in the noncompact case, and [5] in the compact case). Another natural simplification of (1.1) is to impose the Einstein condition on the total space of a Riemannian submersion with totally geodesic fibers [4, 9.61]. One major obstacle in this setup arises from the existence of Yang-Mills connections with curvature form of constant norm. In general, it is still unknown when this necessary condition is satisfied. However, if the structure group of the underlying fiber bundle is a compact torus and the base is closed, then the curvature form of a principal connection is the pullback of a harmonic 2-form on the base. This observation leads people to consider principal torus bundles over products of K¨ ahler-Einstein manifolds and their associated fiber bundles. In these cases, the harmonic 2-forms are rational linear combinations of the Ricci forms. Many interesting Einstein metrics have been discovered on these spaces, among which is the first inhomogeneous Einstein metric with positive scalar curvature constructed by Page [26] on CP 2 ♯ CP 2 , i.e., the nontrivial 2-sphere bundle over CP 1 . Later on, the method of Page was extended by several authors to construct positive Hermitian-Einstein metrics on certain 2-sphere 1991 Mathematics Subject Classification. 53C25. Key words and phrases. Conformally compact Einstein manifold, Q-curvature, Kreck-Stolz invariant. 1