题 目：The stability and uniqueness of N-structures on collapsed manifolds
时 间：2018年6月16日 14:00-16:00
摘 要：The nilpotent Killing structure constructed by Cheeger-Fukaya-Gromov has been a powerful tool in the study of collapsed manifolds with bounded sectional curvature, and draw people's attention in studying collapsed Einstein manifolds. We will talk about the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. We prove that if two metrics on a $m$-manifold of bounded sectional curvature (or bounded Ricci curvature with a positive conjugate radius lower bound) are $L_0$-bi-Lipchitz equivalent and sufficient collapsed (depending on $L_0$ and $m$), then the underlying $N$-structures are isomorphic or one is embedded into another as a subsheaf. It generalizes Cheeger-Fukaya-Gromov's locally compatibility of pure $N$-structures for one collapsed metric of bounded sectional curvature. As an corollary, we prove that those pure $N$-structures constructed by various smoothing method to a bi-Lipschitz equivalent $\epsilon$-collapsed metric of bounded sectional curvature are determined by the original metric uniquely modulo a diffeomorphism.