几何分析讨论班—— Stability of nilpotent structures associated to different collapsed metrics
主 题: 几何分析讨论班—— Stability of nilpotent structures associated to different collapsed metrics
报告人: 胥世成 教授 (首都师范大学)
时 间: 2017-09-28 14:00-15:00
地 点: 理科1号楼1303
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 associated to one fixed metric were established in the work of Cheeger-Fukaya-Gromov and took an essentially role in constructing a global nilpotent structure. 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 would "one fits inside another" as a subsheaf or 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.