Canonical nilpotent structure under bounded Ricci curvature and Reifenberg local covering geometry over regular limits

TL;DR

This paper establishes the existence and uniqueness of a canonical nilpotent structure on collapsed Riemannian manifolds with bounded Ricci curvature, and shows its stability under metric perturbations.

Contribution

It proves the canonical nilpotent structure on regular limit spaces is uniquely determined and equivalent under small metric changes, extending previous results to more general collapsing scenarios.

Findings

Canonical nilpotent structure is uniquely determined up to conjugation.

Nilpotent structures from nearby metrics are equivalent to the canonical one.

The results apply to collapsed manifolds with bounded Ricci curvature and Reifenberg local covering geometry.

Abstract

It is known that a closed collapsed Riemannian $n$-manifold $(M,g)$ of bounded Ricci curvature and Reifenberg local covering geometry admits a nilpotent structure in the sense of Cheeger-Fukaya-Gromov with respect to a smoothed metric $g(t)$. We prove that a canonical nilpotent structure over a regular limit space that describes the collapsing of original metric $g$ can be defined and uniquely determined up to a conjugation, and prove that the nilpotent structures arising from nearby metrics $g_\epsilon$ with respect to $g_\epsilon$'s sectional curvature bound are equivalent to the canonical one.