RCD extension across codimension-two singular sets

Determine whether, for a closed L-infinity Riemannian n-manifold with Ricci curvature bounded below by Kg away from a closed subset of upper Minkowski codimension greater than two, the metric completion of the nonsingular region is an RCD(K,n)-space.

Background

The paper proves an RCD extension theorem when the singular set has upper Minkowski codimension strictly greater than $3-1/(n-1)$, with a slight improvement depending on the bi-Lipschitz comparability constant of the L-infinity metric. The authors then ask whether the codimension threshold can be lowered to the natural value greater than two.

The question is motivated by the sharpness of the codimension-two threshold: the authors note that it cannot be weakened further because two-dimensional cone singularities with sufficiently large cone angle are not RCD spaces. A positive answer would immediately improve the paper’s four-dimensional Einstein removability theorem by replacing its stronger codimension assumption with codimension greater than two.

References

We conclude with a natural open question related to \cref{thm:RCDcodim3}. Let $n 2$ be an integer. Suppose that $(Mn, g)$ is a closed $L\infty$ Riemannian manifold, and $S\subset M$ is a closed subset of upper Minkowski codimension greater than $2$. Suppose that $\mathrm{Ric}(g) Kg$ on $M\setminus S$ for some $K\in\mathbb R$. Is it true that the metric completion of $(M\setminus S, g)$ is an $RCD(K,n)$-space?

— Removability of non-isolated singularities for Einstein metrics and RCD spaces  (2609.01464 - Antonelli et al., 1 Sep 2026) in Question 1.8 (Introduction)