2025/10/30 by Fang, Hanbing, Li, Yu
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.26317
In this paper, we study the singular set 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 S admits a natural stratification: \mathcal S0 ⊂ \mathcal S1 ⊂ ⋯ ⊂ \mathcal Sn-2=\mathcal S, where a point z ∈ \mathcal Sk if and only if no tangent flow at z is (k+1)-symmetric. In general, the Minkowski dimension of \mathcal Sk with respect to the spacetime distance is at most k. We show that the subset Skqc ⊂ Sk, 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 Sk is parabolic k-rectifiable for k ∈ \0, 1, 2\. Furthermore, we establish a sharp uniform \mathscrH2-volume bound for S and show that, up to a set of \mathscrH2-measure zero, the tangent flow at any point in S is backward unique. In addition, we derive L1-curvature bounds for four-dimensional closed Ricci flows.