Special cohomogeneity one metrics with Q^111 or M^110 as principal orbit

TL;DR

This paper classifies cohomogeneity one manifolds with specific principal orbits, constructing metrics with Spin(7) holonomy that are asymptotically conical and analyzing their smoothness at singular orbits.

Contribution

It provides a complete classification of such manifolds with Spin(7) holonomy and constructs explicit metrics with various singular orbits, revealing their asymptotic and smoothness properties.

Findings

Holonomy of constructed metrics is SU(4).

Metrics are asymptotically conical.

Smoothness at singular orbits is analyzed.

Abstract

We classify all cohomogeneity one manifolds with principal orbit Q^111=SU(2)^3/U(1)^2 or M^110=(SU(3) x SU(2))/(SU(2) x U(1)) whose holonomy is contained in Spin(7). Various metrics with different kinds of singular orbits can be constructed by our methods. It turns out that the holonomy of our metrics is automatically SU(4) and that they are asymptotically conical. Moreover, we investigate the smoothness of the metrics at the singular orbit.