Singular sets in noncollapsed Ricci flow limit spaces

TL;DR

This paper provides a detailed stratified description of the singular set in noncollapsed Ricci flow limit spaces, establishing sharp Minkowski-dimension bounds and a parabolic rectifiability theory for strata consisting of cylinder-like singularities. The core approach combines almost splitting maps controlled by entropy pinching with a Reifenberg-type framework and a neck-decomposition scheme, culminating in rectifiability results in all dimensions and sharp 2-dimensional volume bounds for the singular set in dimension four. Key technical advances include a modified entropy for limit spaces, nondegeneration of splitting maps under summability, and a four-dimensional neck decomposition that yields L1-curvature bounds and sharp volume estimates. Together, these results illuminate the precise geometric-measure structure of singularities arising in Ricci flow limits and provide tools for further regularity and stability analysis in geometric evolution problems.

Abstract

In this paper, we study the singular set $\mathcal{S}$ of a noncollapsed Ricci flow limit space, arising as the pointed Gromov--Hausdorff limit of a sequence of closed Ricci flows with uniformly bounded entropy. The singular set $\mathcal{S}$ admits a natural stratification: \begin{equation*} \mathcal S^0 \subset \mathcal S^1 \subset \cdots \subset \mathcal S^{n-2}=\mathcal S, \end{equation*} where a point $z \in \mathcal S^k$ if and only if no tangent flow at $z$ is $(k+1)$-symmetric. In general, the Minkowski dimension of $\mathcal S^k$ with respect to the spacetime distance is at most $k$. We show that the subset $\mathcal{S}^k_{\mathrm{qc}} \subset \mathcal{S}^k$, consisting of points where some tangent flow is given by a standard cylinder or its quotient, is parabolic $k$-rectifiable. In dimension four, we prove the stronger statement that each stratum $\mathcal{S}^k$ is parabolic $k$-rectifiable for $k \in \{0, 1, 2\}$. Furthermore, we establish a sharp uniform $\mathscr{H}^2$-volume bound for $\mathcal{S}$ and show that, up to a set of $\mathscr{H}^2$-measure zero, the tangent flow at any point in $\mathcal{S}$ is backward unique. In addition, we derive $L^1$-curvature bounds for four-dimensional closed Ricci flows.

Theorems & Definitions (220)

• Theorem 1.1
• Definition 1.2: Parabolic $k$-rectifiability
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 1.10: $(\delta,r)$-selfsimilar