Dice Question Streamline Icon: https://streamlinehq.com

Existence of RCD(2,3) metrics on S^3/Γ producing Ricci cone limits

Determine whether, for any 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 cone C(S^3/Γ) is a noncollapsed Ricci limit space.

Information Square Streamline Icon: https://streamlinehq.com

Background

This question asks for the smoothability and realizability of cones over spherical space forms within the noncollapsed Ricci limit class. Kronheimer’s work shows many Ricci‑flat 4‑manifolds have cones at infinity modeled on C(S3/Γ), but whether such cones arise as limits from the lower Ricci curvature side via suitable RCD(2,3) metrics on the base remains unclear.

A positive answer would provide concrete families of singular Ricci limit spaces modeled on cones over 3D space forms, informing the landscape of allowable tangent cone cross‑sections.

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 Section 1.6 (Question 1.15)