Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three

Starting from a closed, orientable three-dimensional Riemannian manifold, we consider the completion of the associated singular Ricci flow with respect to a natural spacetime distance. We show that this completion admits canonical intrinsic metrics on its time-slices, defined by conjugate heat kernel measures, and that these time-slices arise as metric limits of the regular part of the flow. More precisely, for any $t_0>0$ and any connected component $Z_{t_0}^{\mathrm{\prime }}$ of the time-slice $Z_{t_0}$ of the completion, we prove that $$(\overline{{\mathcal{R}}_t^{\mathrm{\prime }}},d_{g_t})\underset{t\nearrow t_0}{\overset{\mathrm{G}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{v}\text{-}\mathrm{H}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{d}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{f}}{\to }}(Z_{t_0}^{\mathrm{\prime }},d_{t_0}^Z),$$ where ${\mathcal{R}}_t^{\mathrm{\prime }}$ is the corresponding connected component of the regular part and $\overline{{\mathcal{R}}_t^{\mathrm{\prime }}}$ denotes its metric completion. In particular, this yields the Gromov-Hausdorff convergence at the first singular time for closed three-dimensional Ricci flows. We also establish a refined structure theory for the singular set of the completion. In particular, the singular set is horizontally parabolic $1$-rectifiable, and its time image has vanishing $1\mathrm{/}2$-dimensional Hausdorff measure. Moreover, on each time-slice, the singular set has Minkowski dimension at most $1$. The proof relies on heat kernel estimates for singular Ricci flows and the generalization of the structure theory of noncollapsed Ricci flow limit spaces.