New Infinite Series of Einstein Metrics on Sphere Bundles from AdS Black Holes

TL;DR

The paper constructs new infinite families of Einstein metrics with positive scalar curvature on sphere bundles by taking a Page-type limit of nearly extreme AdS Kerr black holes in varying dimensions. The 5D case yields explicit inhomogeneous metrics on $S^3$-bundles over $S^2$ and homogeneous ones when the bundle data align, with the geometry controlled by parameters $(\nu_1,\nu_2)$ and integers $(k_1,k_2)$. The higher-dimensional generalization produces a one-parameter family of Einstein metrics on the nontrivial $S^{d-2}$-bundle over $S^2$ for $d\ge4$, recovering the Page metric at $d=4$ and yielding cohomogeneity-one examples with principal orbits $S^3\times S^{d-4}$ for other cases. These constructions extend Page’s method and connect to Wang–Ziller homogeneous metrics, enriching the catalog of explicit Einstein metrics and highlighting the role of AdS black hole limits in geometric analysis.

Abstract

A new infinite series of Einstein metrics is constructed explicitly on S^2 x S^3, and the non-trivial S^3-bundle over S^2, containing infinite numbers of inhomogeneous ones. They appear as a certain limit of a nearly extreme 5-dimensional AdS Kerr black hole. In the special case, the metrics reduce to the homogeneous Einstein metrics studied by Wang and Ziller. We also construct an inhomogeneous Einstein metric on the non-trivial S^{d-2}-bundle over S^2 from a d-dimensional AdS Kerr black hole. Our construction is a higher dimensional version of the method of Page, which gave an inhomogeneous Einstein metric on CP^2\sharp\bar{CP^2}.