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arXiv · 2609.33995

Codimension-four regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound

Lingling Kong
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Abstract

Let $(X,d,p)$ be an $n$-dimensional noncollapsed Ricci limit space, where $n\ge4$. Under an integral volume-deficit bound, we prove that the metric singular set has Hausdorff codimension at least four and sigma-finite $(n-4)$-dimensional Hausdorff measure. This establishes a special case of the codimension-four regularity conjecture. Moreover, the nonmanifold locus is closed and has locally finite $(n-4)$-dimensional Hausdorff measure, and its intersection with each bounded ball satisfies a tubular-volume estimate of order $r^4$. In dimension four, the nonmanifold points form a locally finite set. A flat quotient example shows that the codimension bound is sharp.

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Lingling Kong. 2026-09-27. Codimension-four regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound. https://arxiv.org/abs/2609.33995

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