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

BiHölder smoothability for noncollapsed RCD(−2,3) spaces with Euclidean tangents

Show that any noncollapsed RCD(−2,3) space in which all tangent cones are Euclidean is biHölder homeomorphic to a smooth Riemannian 3‑manifold.

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

Background

The paper proves topological manifold recognition under Euclidean tangent cone assumptions (Theorem 1.8) but only establishes a topological, not smooth, structure. The conjecture strengthens this to a biHölder equivalence with smooth manifolds, paralleling results known for Ricci limit spaces via Ricci flow in three dimensions.

A solution would align synthetic RCD regularity with smooth geometric regularity, enhancing the quantitative control of the metric to a biHölder equivalence with smooth structures.

References

Conjecture 1.17. Any noncollapsed RCD(−2,3) space with Euclidean tangent cones is biHölder homeomorphic to a smooth Riemannian manifold.

— Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below (2405.03839 - Bruè et al., 6 May 2024) in Section 1.6 (Conjecture 1.17)