Speaker：Xiaochun Rong (Rutgers University)

Time：2023-7-19, 9:30

Location：Conference Room 105 at Experiment Building at Haiyun Campus

Abstract:

A complete Riemannian n-manifold M is called epsilon-collapsed, if every unit ball in M has volume less than epsilon (while often a bound on `curvature' must be imposed to prevent a rescaling of metric). In 1978, Gromov classified `almost flat manifolds' (or the `maximally collapsed manifolds' with sectional curvature bounded in absolute value by one and small diameter) ; a bounded normal covering space of M is diffeomorphic to the quotient of a simply connected nilpotent Lie group modulo a manifold up to a co-compact lattice. This result has been a corner stone in the collapsing theory of Cheeger-Fukaya-Gromov in 90's that there is a nilpotent structure on any epsilon-collapsed manifold with bounded sectional curvature, and this theory has found important applications in Metric Riemannian geometry.

We will survey some recent development in generalizing the collapsing theory to epsilon-collapsed manifolds of Ricci curvature bounded below and the (incomplete) universal cover of every unit ball in M is not collapsed. The study of this class of collapsed manifolds is partially fueled with many constructions of collapsed Calabi-Yau metrics using certain underlying singular nilpotent fibrations.