On the collapsing of spaces with local rewinding Reifenberg control

Let $(M^n,g_0)$ be a complete Riemannian manifold with ${\text{Ric}}_{g_0}\ge -(n-1)g_0$ whose local universal covers are uniformly $\delta$-Reifenberg-close to ${\mathbb{R}}^n$ below a fixed radius $\rho$. We show that for every $\alpha \in (0,1)$ there are $\delta (n,\alpha )>0$ and $T(n,\rho ,\alpha )>0$ such that, if $\delta \le \delta (n,\alpha )$, a complete Ricci flow $g(t)$ with $g(0)=g_0$ and $\mathrm{\mid }{\text{Rm}}_{g(t)}\mathrm{\mid }\le \alpha \mathrm{/}t$ exists on $(0,T]$. On closed manifolds this is a uniform lower bound for the lifetime of the Ricci flow; on noncompact manifolds, where the initial curvature may be unbounded, it is an existence theorem, and the flow is unique among complete Ricci flows from $g_0$ with a curvature bound of the form $A\mathrm{/}t$. The key step is a finite-buffer comparison of two Ricci flows through the Ricci–harmonic map heat flow, following Lee, which transfers a Type-I curvature bound from a complete selected flow to the incomplete flow lifted to a local cover. Consequently, below a dimensional Reifenberg threshold, collapsing closed manifolds admit singular nilpotent fibrations over their Gromov–Hausdorff limits. The Reifenberg hypothesis cannot be weakened to noncollapsing of the local covers: products of shrinking circles with Nifa’s four-manifolds have $\text{Ric}\ge 0$ and uniformly noncollapsed local covers, but admit no continuous Gromov–Hausdorff approximations onto their limit.