Compact Einstein Spaces based on Quaternionic K\"ahler Manifolds

TL;DR

This paper constructs new inhomogeneous compact Einstein spaces with orbifold singularities by reducing Einstein equations to ODEs on bundles over Quaternionic Kähler manifolds, using numerical methods.

Contribution

It introduces a novel approach to find Einstein metrics on bundles over Quaternionic Kähler manifolds, leading to inhomogeneous solutions with orbifold singularities.

Findings

Numerical solutions of Einstein equations on specific bundles.

Existence of inhomogeneous compact Einstein spaces with orbifold singularities.

Reduction of Einstein equations to a manageable set of ODEs.

Abstract

We investigate the Einstein equation with a positive cosmological constant for $4n+4$-dimensional metrics on bundles over Quaternionic K\"ahler base manifolds whose fibers are 4-dimensional Bianchi IX manifolds. The Einstein equations are reduced to a set of non-linear ordinary differential equations. We numerically find inhomogeneous compact Einstein spaces with orbifold singularity.