On the collapsing of spaces with local rewinding Reifenberg control

Abstract

Let $(M^n,g_0)$ be a complete Riemannian manifold with $\Ric_{g_0}\ge-(n-1)g_0$ whose local universal covers are uniformly $δ$-Reifenberg-close to $\R^n$ below a fixed radius $ρ$. We show that for every $α\in(0,1)$ there are $δ(n,α)>0$ and $T(n,ρ,α)>0$ such that, if $δ\leδ(n,α)$, a complete Ricci flow $g(t)$ with $g(0)=g_0$ and $|\Rm_{g(t)}|\leα/t$ 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 $A/t$. 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 $\Ric\ge0$ and uniformly noncollapsed local covers, but admit no continuous Gromov--Hausdorff approximations onto their limit.