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Existence of RCD(2,3) metrics on spherical space forms producing Ricci-limit cones

Determine whether, for each discrete group Γ < O(4) acting freely on S^3, there exists an RCD(2,3) metric on the spherical space form S^3/Γ such that the metric cone C(S^3/Γ) is realized as a noncollapsed Ricci limit space.

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Background

This question probes the smoothability and realizability of metric cones over spherical space forms within the class of noncollapsed Ricci limit spaces. While certain ALE spaces and cones (e.g., over the Poincaré homology sphere) arise from Ricci-flat metrics, a general existence result for arbitrary free Γ-actions on S3 is unknown.

An affirmative answer would tie together RCD regularity, orbifold metrics, and the structure of cones appearing as limits of smooth manifolds with Ricci curvature bounds.

References

Question 1.15. Let Γ < O(4) be a discrete group acting freely on S . Is there an RCD(2,3) metric over S /Γ such that C(S /Γ) is 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 1.15, Section 1.6 (Open questions)