Student Geometry Seminar

Tuesdays, 4-5pm, Evans 732

Schedule of Talks

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 15 TBA TBA
Nov 22 TBA TBA
Nov 29 TBA TBA