2023年冬短期课程-Quantitative Maximal Rigidity of Ricci Curvature Bounded Below
Quantitative Maximal Rigidity of Ricci Curvature Bounded Below
A goal of this mini-course is to give a quick introduction to an on-going research area (in the title) in Metric Riemannian geometry.
In Riemannian geometry, a maximal rigidity on an n-manifold M of Ricci curvature bounded below by (n−1)H is a statement that a geometric or a topological quantity of M is bounded above by that of an n-manifold of constant sectional curvature H, and “=” implies that M has constant sectional curvature H.
A quantitative maximal rigidity of Ricci curvature bounded below is a statement that if a geometric quantity is almost maximal, then M admits a nearby metric of constant sectional curvature H (which may require additional conditions). Indeed, the Cheeger-Colding-Naber theory on Ricci limit spaces initiated in establishing a quantitative maximal volume rigidity of positive Ricci curvature.
1. Maximal rigidities of Ricci curvature bounded below
2. Gromov-Hausdorff topology, and Ricci limit spaces
3. Quantitative maximal rigidities of Ricci curvature bounded below
4. Structures on a Ricci limit space of a collapsing sequence with local Ricci bounded covering geometry