Singular sets in noncollapsed Ricci flow limit spaces

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: \beginequation* \mathcal S^0 \subset \mathcal S^1 \subset \cdots \subset \mathcal S^n-2=\mathcal S, \endequation* where a point $z\in {\mathcal{S}}^k$ if and only if no tangent flow at $z$ is $(k+1)$-symmetric. In general, the Hausdorff dimension of ${\mathcal{S}}^k$ with respect to the spacetime distance is at most $k$. We show that the subset ${\mathcal{S}}_{\mathrm{qc}}^k\subset {\mathcal{S}}^k$, 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 ${\mathcal{H}}^2$-volume bound for $\mathcal{S}$ and show that, up to a set of ${\mathcal{H}}^2$-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. As an application, we resolve Perelman's bounded diameter conjecture for three-dimensional closed Ricci flows.