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Smoothability of noncollapsed RCD(−2,3) 3-manifolds as Ricci limits

Ascertain whether every noncollapsed RCD(−2,3) metric measure space (X^3,d,H^3) that is a topological 3-manifold is a noncollapsed Ricci limit space.

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Background

The RCD framework generalizes lower Ricci bounds to nonsmooth metric measure spaces. The authors' manifold recognition theorem (Theorem 1.8) characterizes when three-dimensional RCD spaces are manifolds, but whether all such noncollapsed RCD(−2,3) manifolds arise as limits of smooth Ricci-bounded manifolds remains unknown.

This is a central smoothability question analogous to longstanding open problems for Alexandrov spaces.

References

Question 1.16. Let (X 3,d) be a noncollapsed RCD(−2,3) spaces. Assume that X is a topological manifold. Is (X ,d) a noncollapsed Ricci limit space?

— Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below (2405.03839 - Bruè et al., 6 May 2024) in Question 1.16, Section 1.6 (Open questions)