Naber’s k‑symmetry conjecture for tangent cones of noncollapsed Ricci limits

Show that in any noncollapsed n‑dimensional Ricci limit space (X,d), at each point all tangent cones are k‑symmetric outside a subset of Hausdorff dimension less than k−1.

Background

The conjecture provides quantitative regularity of tangent cones, predicting that at most scales and points the tangent cones split large Euclidean factors, except on sets of controlled dimension. Combined with Theorem 1.4(ii) in this paper, such a result would settle the topological stability part of [41, Conjecture 1.2].

It is a central structural assertion in the paper of singularities of noncollapsed Ricci limit spaces, aiming to refine the stratification by symmetry and improve control over tangent cone variation.

References

A conjecture by Naber [ 84, Conjecture 2.16] predicts that for a noncollapsed n Ricci limit space (X ,d) all tangent cones at a given point should be k-symmetric away from a set of Hausdorff dimension less than k − 1.

— Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below (2405.03839 - Bruè et al., 6 May 2024) in Remark 1.5