Stability of Nilpotent Structures of Collapsed Manifolds on the Same Scale
Abstract：We will talk about a recent work on the stability of nilpotent structures on a collapsed manifold with bounded sectional or Ricci curvature. A manifold of bounded sectional curvature is called epsilon-collapsed, if the injectivity radius, or equivalently the volume of unit ball, at every points is less than epsilon. The geometry/topology of a collapsed manifold can be totally described by Cheeger-Fukaya-Gromov's nilpotent structure. Similar results had been extended to manifolds of bounded Ricci curvature under some additional assumptions. The stability of locally defined nilpotent structures were essential in the work of Cheeger-Fukaya-Gromov to construct a global nilpotent structure on one fixed metric. Nilpotent structures also depend on the choice of epsilon, the collapsed length scale one inspects. We prove that if two metrics on a manifold are L_0-bi-Lipchitz equivalent and sufficient collapsed (depending on L_0) under bounded (Ricci) curvature, then the underlying nilpotent structures are isomorphic to each other. As applications, we establish a link between the components of the moduli space of all collapsed Riemannian metrics and the set of isomorphism classes of nilpotent structures,and derive a new parametrized version of Gromov's flat manifold theorem under bounded Ricci curvature and conjugate radius.