Spin(7) metrics of cohomogeneity one with Aloff--Wallach spaces as principal orbits

TL;DR

This paper constructs new families of non-compact Spin(7) metrics with Aloff--Wallach spaces as principal orbits, including transitions between different asymptotic geometries, enriching the understanding of special holonomy manifolds.

Contribution

It introduces two continuous families of Spin(7) metrics with Aloff--Wallach spaces as principal orbits, including geometric transitions for the case N_{1,1}.

Findings

Metrics are locally asymptotically conical (ALC) for generic Aloff--Wallach spaces.

For N_{1,1}, metrics exhibit a transition from asymptotically conical to ALC.

Constructed metrics have both chiralities of Spin(7).

Abstract

In this article, we construct two continuous 1-parameter family of non-compact $\mathrm{Spin}(7)$ metrics with both chiralities, with the principal orbit an Aloff--Wallach space $N_{k,l}$ and the singular orbit $\mathbb{CP}^2$. For generic $N_{k,l}$, metrics constructed are locally asymptotically conical (ALC). For $N_{1,1}$, we construct two continuous 1-parameter families with geometric transition from asymptotically conical (AC) metrics to ALC metrics.