Spring 2022: The tensor maximum principle and applications
Meeting time: Wednesdays 10:30am - 11:30am
| Topic | Reference | Speaker | Date & Location |
|---|---|---|---|
| Matrix maximum principle; Hamilton’s matrix Harnack inequality |
[L] §10.2-10.3 | Julius | Feb 16, 2-429 |
| Statement of vector bundle weak & strong maximum principle; Special cases, examples; Support function of a convex set |
[S] §4.5 | Julius | Feb 23, 2-429 |
| Viscosity solutions; Scalar maximum principle for viscosity solutions; Proof of vector bundle maximum principle (weak & strong) |
[S] §4.4-4.5 | Tang-Kai | Mar 2, 2-361 |
| Preservation of PIC along Ricci flow | [B2] Ch. 7 | Xinrui | Mar 9, 2-361 |
| Preservation of PIC along Ricci flow | [B2] Ch. 7 | Xinrui | Mar 16, 2-361 |
| Brendle’s generalization of Hamilton’s Harnack inequality for RF | [B1] | Keaton | Mar 23, via Zoom |
| PIC+ in 4 dimensions | [RS], [F] | Tristan | Mar 30, via Zoom |
| Huisken's MCF convexity theorem | [Hu] | Qin | Apr 6, 2-449 |
| Hamilton's Harnack estimate for MCF | [Ha] | Tang-Kai | Apr 13, 2-361 |
| Improved Hamilton Harnack estimate (for MCFs without bounded curvature assumption) | [DS] | Keaton | Apr 20, 2-361 |
| The nature of singularities in mean curvature flow of mean-convex sets | [W] | Feng | Apr 27, 2-361 |
| References | |
|---|---|
| [B1] | Brendle, S. (2009). A generalization of Hamilton's differential Harnack inequality for the Ricci flow. Journal of Differential Geometry, 82(1), 207-227. |
| [B2] | Brendle, S. (2010). Ricci flow and the sphere theorem (Vol. 111). American Mathematical Soc. |
| [DS] | Daskalopoulos, P., & Saez, M. (2021). Uniqueness of entire graphs evolving by Mean Curvature flow. arXiv preprint arXiv:2110.12026. |
| [F] | Fine, J. (2011). A gauge theoretic approach to the anti-self-dual Einstein equations. arXiv preprint arXiv:1111.5005. |
| [Ha] | Hamilton, R. S. (1995). Harnack estimate for the mean curvature flow. Journal of Differential Geometry, 41(1), 215-226. |
| [Hu] | Huisken, G. (1984). Flow by mean curvature of convex surfaces into spheres. Journal of Differential Geometry, 20(1), 237-266. |
| [L] | Lee, T.K. (2021). Topics in the heat equation (18.966 lecture notes). (link) |
| [RS] | Richard, T., & Seshadri, H. (2016). Positive isotropic curvature and self-duality in dimension 4. manuscripta mathematica, 149(3), 443-457. |
| [S] | Sun, A. (2020). Lecture notes on Ricci flow (taughty by Richard Bamler). (link) |
| [W] | White, B. (2003). The nature of singularities in mean curvature flow of mean-convex sets. Journal of the American Mathematical Society, 16(1), 123-138. |