Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

TL;DR

This paper classifies and constructs new cohomogeneity-one Einstein and Spin(7) holonomy metrics on Aloff-Wallach spaces, using power series methods to analyze singular orbits and convergence.

Contribution

It provides the first classification results for such metrics near singular orbits and introduces new Einstein metrics with special holonomy constructed from Aloff-Wallach spaces.

Findings

Classified metrics near various singular orbits.

Discovered many new Einstein metrics of cohomogeneity one.

Applied power series techniques for convergence and smoothness proofs.

Abstract

We investigate cohomogeneity-one metrics whose principal orbit is an Aloff-Wallach space SU(3)/U(1). In particular, we are interested in metrics whose holonomy is contained in Spin(7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metrics on tubular neighborhoods of various singular orbits. Since the equation for the holonomy reduction has only few explicit solutions, we make use of power series techniques. In order to prove the convergence and the smoothness near the singular orbit, we apply methods developed by Eschenburg and Wang. As a by-product of these methods, we find many new examples of Einstein metrics of cohomogeneity one.