Cohomogeneity one $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$ metrics with generic Aloff--Wallach spaces as principal orbits

This paper establishes the existence of forward complete cohomogeneity one $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$ metrics with generic Aloff–Wallach spaces $N_{k,l}$ as principal orbits and ${\mathbb{C}\mathbb{P}}^2$ as the singular orbit, building on Reidegeld’s analysis of the initial value problem. We construct three continuous one-parameter families of non-compact $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$ metrics. Each family contains a limiting asymptotically conical (AC) metric, while the other metrics in the families are asymptotically locally conical (ALC). Moreover, two of the AC metrics share the same asymptotic cone, exhibiting a geometric transition phenomenon analogous to that found by Lehmann in the exceptional case.