Title：Stability of Nilpotent Structures of Collapsed Manifolds on the Same Scale
Speaker：胥世成（首都师范大学）
Time：2017年9月22日    下午4:00--5:30
Room：东区管理科研楼  数学科学学院1418室
Abstract：We will talk about arecent work on the stability of nilpotent structures on a collapsed manifoldwith bounded sectional or Riccicurvature. A manifold of bounded sectionalcurvature is called /epsilon-collapsed, if the injectivity radius, or equivalentlythe volume of unit ball, at every points is less than /epsilon.Thegeometry/topology of a collapsed manifold can be totally described byCheeger-Fukaya-Gromov's nilpotent structure. Similar results had been extendedto manifolds of bounded Riccicurvature under some additional assumptions. Thestability of locally defined nilpotent structures were essential in the work ofCheeger-Fukaya-Gromov to construct a global nilpotent structure onone fixedmetric. Nilpotent structures also depend on the choice of /epsilon,thecollapsed length scale one inspects. We prove that if two metrics on amanifoldare L_0-bi-Lipchitz equivalent and sufficient collapsed (depending onL_0) underbounded (Ricci) curvature, then the underlying nilpotent structuresareisomorphic to each other.  Asapplications,we establish a link between the components of the moduli space ofall collapsedRiemannian metrics and the set of isomorphism classes ofnilpotentstructures,and derive a new parametrized version of Gromov's flatmanifoldtheorem under bounded Ricci curvature and conjugate radius.