Collapsed Manifolds of Bounded Curvature
A Riemannian n-manifold M is called ϵ-collapsed, if every unit ball on M has a volume < ϵ(n) (a small constant depending on n), while a suitable bound on curvature is imposed in preventing a possible scaling.
This graduate topic course consists of two parts: In the first part, we will start with the Gromov’s theorem on almost flat manifolds, then cover basics of the nilpotent structure theory of Cheeger-Fukaya-Gromov on collapsed n-manifolds with sectional curvature bounded in absolute value, as well as applications. In the second part of this course, we will present partial nilpotent structures on collapsed n-manifolds with Ricci curvature bounded in absolute value, or bounded below, under additional restrictions.
CONTENTS
Part I. Collapsed Manifolds with Sectional Curvature Bounded in Absolute Value
0. Preliminaries
1. The Gromov-Hausdorff topology and equivariant GH-convergence
2. Fiber bundle theorems
3. The stability of compact Lie group actions
4. Gromov’s almost Flat Manifolds 5. The Singular nilpotent fibration
Part II. Collapsed Manifolds with Ricci Curvature Bounded Below and Local Bounded Covering Geometry
6. Basics in the theory of Ricci limit spaces by Cheeger-Colding-Naber
7. Nilpotent structures on collapsed manifolds with local Ricci bounded covering geometry
8. Application: quantitative maximal rigidity of Ricci curvature bounded below
The instructor prepares to write lecture notes for this course.
A basic knowledge in Riemannian geometry is required for this course, we will try to make the topics self-contained and accessible to some early graduate students.