Math 18.966: Geometry of Manifolds II
Tuesday, Thursday, 9:30-11am, 2-147
Course website:
math.mit.edu/~tristanc/Math18.966.html
Instructor:
Tristan Collins
Email: tristanc at math dot mit dot edu
Office: 2-273
About the course
This course will cover aspects of comparison geometry, Ricci curvature, and the convergence theory of Riemannian manifolds as well as some applications.
Syllabus
The syllabus for the course is available
here .
Course References
As much as possible, I will try to provide links to publicly available (read arXiv) sources. If possible, I encourage you to seek out the published versions of these papers. Many textbooks are available on the web via the MIT Library.
- (K) Karcher: Riemannian Comparison Constructions (PDF)
- (Sz) Székelyhidi: Introduction to Extremal metrics (PDF)
- (Ch) Cheeger: Degeneration of Riemannian Metrics Under Ricci Curvature Bounds
- (Pet) Petersen: Riemannian Geometry
- (Ev) Evans: Partial Differential Equations
- (GT) Gilbarg and Trudinger: Elliptic Partial Differential Equations of Second Order
- (LH) Lin and Han: Elliptic Partial Differential Equations
- (An) Anderson: Convergence and rigidity of manifolds under Ricci curvature bounds
Progress
This is approximate. I will attempt to keep track of all the references used in class.
| Date |
Material |
Problems/Handouts |
| Feb.5--26 |
Geometry of generalized distance functions: References (K)
|
Homework 1
|
| Feb. 28--Mar. 14 |
Toponogov's Theorem, Gromov-Hausdorff Convergence: References (K), (Pet) |
Homework 2
|
| Mar. 19- Apr. 5 |
Elliptic PDE: The maximum princple, regularity in Sobolev spaces, Schauder theory (Ev), (GT), (LH), (Sz). Applications ot Hodge theory (Pet) |
Homework 3
|
| Apr. 9- 11 |
Cheeger-Gromoll splitting (Ch) |
|
Apr. 16-23 |
Harmonic coordinates Existence of Harmonic coordinates, Fundamental compactness theorem of Riemannian manifolds (Pet) |
|
| Apr. 25-- |
Applications Anderson's estimate for Harmonic radius (An), Cheeger's finiteness theorem (Pet) |
Homework 4
|
