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 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.

26 pages