Pan-style collapsing constructions for α‑Grushin spaces beyond α=1 and identification of the limiting measure

Determine whether, for α ≠ 1, there exist sequences of doubly warped product metric spaces with positive Ricci curvature whose collapsed limits yield α‑Grushin half-spaces (such as the α‑Grushin hemisphere S^+_α), and ascertain whether the limiting measure in such constructions coincides with the β‑weighted measure employed in this paper (for S^+_α, m^β_{S_α} = cos(x) sin^{β−α}(x) dx dy).

Background

Pan (2023) constructed a sequence of doubly warped product Riemannian manifolds whose collapsed Ricci limit space satisfies RCD(0,n+1) and was identified as the 1‑Grushin hemisphere. The authors generalize α‑Grushin constructions and introduce β‑weighted measures tailored to obtain RCD(K,N) across new models.

It remains unknown whether analogous collapsing sequences can be built for α‑Grushin spaces with α ≠ 1 and whether the resulting limit measure matches the β‑weighted measures defined for these models, which would provide a unified geometric and measured limit framework across the α‑Grushin family.