Sep 13 | Alex Appleton | Paper: H. Koch, T. Lamm, Geometric flows with rough initial data (2009)
(arxiv) Abstract of paper: We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in \(L^\infty\) and to the harmonic map flow for initial maps whose image is contained in a small geodesic ball. |
Sep 20 | Alex Appleton | Continuation of the previous week |
Sep 27 | Michael Smith |
Title: Local \(L^{1/2-\epsilon}\) bound on Ricci curvature assuming Ricci curvature bounded from below Abstract: Assuming a lower bound on the Ricci curvature of a Riemannian manifold, we show that the Ricci curvature along a minimizing geodesic of length less than or equal to 2 is bounded in \(L^{1/2-\epsilon}\), by an argument using the second variation formula and some simple inequalities. we also show that the volume of a 1-ball and the amount, in some sense, of minimizing geodesics of length \(l \) satisfying \( 1 < l < 2 \) emanating from the ball are related. In particular, small volume implies few of these geodesics and conversely, a lower bound on the volume implies a lower bound on the amount of such geodesics. We use these relations and the Ricci curvature bound along minimizing geodesics to achieve a similar bound for the Ricci curvature within the ball. |
Oct 4 | Patrick Wilson (maybe one week later) |
Paper: A. Deruelle, T. Lamm, Weak stability of Ricci expanders with positive curvature operator, (2016)
(arxiv) Abstract of the paper: We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity. |
Oct 11 | TBA | TBA |
Oct 18 | TBA | TBA |
Oct 25 | TBA | TBA |
Nov 1 Evans 961 | TBA | TBA |
Nov 8 | TBA | TBA |
Nov 15 | TBA | TBA |
Nov 22 | TBA | TBA |
Nov 29 | TBA | TBA |