The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers

Abstract

Let compact $\mathrm{RCD}(\kappa ,N)$ spaces converge in the measured Gromov--Hausdorff sense, and suppose that they have $(\delta ,\rho )$-Reifenberg local covers for one fixed $\rho >0$, with $|\kappa |{\rho }^2\le 1$ and $\delta$ small. We prove that, along a subsequence, the spaces are singular fibrations over the limit: their canonical frame spaces converge to a topological manifold with a locally linear action of $O(n)$ whose quotient is the limit; for large indices the frame spaces fibre over this manifold, by maps that commute with $O(n)$, with fibres homeomorphic to nilmanifolds; and the induced maps to the limit are, near every point, quotients of products by the isotropy group, with fibres homeomorphic to infranilmanifolds. As applications, we extend a fibration theorem of Honda and Zhang to non-smooth RCD spaces whose local covers have injectivity radius bounded below, and we show that for aspherical spaces the isotropy groups are finite, the fibre groups inject into the fundamental groups, and the limit is the quotient of a contractible manifold by a proper cocompact action; under Rong's condition $A_s$ it is a closed aspherical manifold. Three appendices treat smooth manifolds with $\mathrm{Ric}\ge -(n-1)$ and small local rewinding Reifenberg error: a uniform short-time Ricci flow with $|\mathrm{Rm}|\le \alpha /t$, also for complete non-compact data, where it is unique among complete flows with curvature at most $A/t$; a second construction with affine nilmanifold bundles; and, conditional on Nifa's construction, examples showing that rewinding volume control alone does not give continuous Gromov--Hausdorff approximations with small error.

arXiv commentUpdated version of the original paper titled "On the collapsing of spaces with local rewinding Reifenberg control" with new results along the same line of research; no separate submission of the results due to arxiv monthly submission restriction