Examples of Einstein manifolds in odd dimensions

TL;DR

This paper constructs and classifies various Einstein metrics on odd-dimensional manifolds, revealing new examples with specific curvature and topological properties, including conformally compact and Ricci-flat metrics.

Contribution

It introduces new Einstein metrics on solid torus and 3-sphere bundles over Fano Kahler-Einstein manifolds, expanding the known landscape of Einstein manifolds in odd dimensions.

Findings

Negative Einstein metrics are conformally compact.

Ricci-flat metrics exhibit slower-than-Euclidean volume growth.

Classified topological types of total spaces over complex projective plane.

Abstract

We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler-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 Kahler-Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base is the complex projective plane.