Einstein Metrics on Spheres

TL;DR

This paper demonstrates the existence of numerous new Einstein metrics on odd-dimensional spheres, including exotic spheres, with the number of solutions growing double exponentially with dimension, using orbifold and singularity techniques.

Contribution

It introduces a novel method to construct Einstein metrics on spheres via Brieskorn-Pham singularities and orbifold lifts, expanding the known family of such metrics.

Findings

Countless new Einstein metrics on spheres including exotic ones.

Number of metric families grows double exponentially with dimension.

Method leverages complex algebraic orbifolds and singularities.

Abstract

We prove the existence of an abundance of new Einstein metrics on odd dimensional spheres including exotic spheres, many of them depending on continuous parameters. The number of families as well as the number of parameter grows double exponentially with the dimension. Our method of proof uses Brieskorn-Pham singularities to realize spheres (and exotic spheres) as circle orbi-bundles over complex algebraic orbifolds, and lift a Kaehler-Einstein metric from the orbifold to a Sasakian-Einstein metric on the sphere.