--- This summer is about the convergence of Riemannian manifolds --- if you are in Toronto I'll treat you with summerlicious after every July meeting.
We will consider Riemannian manifolds with uniformly prescribed conditions on sectional curvature, Ricci curvature or scalar curvature, as well as other metric geometry conditions (e.g. volume, diameter, measure), and try to understand the limiting behavior of a family of such Riemannian manifolds.
Zoom online meeting on 3 PM (PDT), Wednesdays.
May 26 (Junsheng): Basic comparison geometry from the Weitzenbock formula, and segment inequality. ([Zhu], [Cheeger])
June 2 (Holly): Volume comparison, Gromov compactness, and fundamental groups. ([Zhu])
June 9 (Charlie): Laplace comparison, Cheeger-Gromoll splitting, and Abresch-Gromoll excess estimate. ([Zhu])
June 16 (Mingyang): Anderson's convergence under bounded Ricci curvature. ([An90])
June 23 (Hongyi): Quantitative maximum principle, gradient estimate and a good cut-off function. ([Cheeger])
June 30 (Junsheng): Quantitative splitting theorem and volume convergence. ([Cheeger])
July 7 (Holly): Volume cone implies metric cone. ([Cheeger])
July 14 (Charlie): Stratification of Ricci limit spaces. ([Cheeger])
July 21 (Hongyi): Heat kernel smoothing of distance function. ([CN12])
July 28 (Mingyang): Gromov-Hausdorff closeness of balls along a geodesic with nearby centers. ([CN12])
[Zhu] Shunhui Zhu, The comparison geometry of Ricci curvature.
[An90] Michael Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds.
[Cheeger] Jeff Cheeger, Degeneration of Riemannian metrics under Ricci curvature bounds.
[CN12] Tobias Colding and Aaron Naber, Sharp Holder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications.
[An89] Michael Anderson, Curvature bounds and Einstein metrics on compact manifolds.
[An92] Michael Anderson, The L^2 structure of moduli spaces of Einstein metrics on 4-manifolds.
[TV] Gang Tian and Jeff Viaclovsky, Bach-flat asymptotically locally Euclidean metrics.
[F87a] Kenji Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator.
[F87b] Kenji Fukaya, Collapsing Riemannian manifolds to ones of lower dimensions.
[F88] Kenji Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvature and diameter.