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

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 study the nilpotent structures over a regular limit space with optimal regularities that describe the collapsing of original metric $g$, and prove that they are uniquely determined up to a conjugation by diffeomorphisms with bi-Lipschitz constant almost $1$, and are equivalent to nilpotent structures arising from other nearby metrics $g_{𝜖}$ with respect to $g_{𝜖}$’s sectional curvature bound.