Course Category：Advanced Course
This is an intermediate course on geometric analysis. Geometric analysis is a very active subject, which is very successful in the last few decades. It is impossible to cover it in a short course like this. We have to make a subjective choice of topics to present so that the audience will have a taste of the whole subject. Unfortunately, this means that there is many other important aspects will not be covered in this short course.
We outline an ambitious plan below. We may skip one or two topics in this list to better cover the remaining topics.
More specifically, we will cover the following four topics：
1.Re-visit of basic Riemannian geometry: metric, geodesic, curvature and Jacobi vector fields (various estimates). 1st and 2nd variation of length function. Classical comparison theorem like Toponogov, Rauch comparison.
2.Introduction of geometric analysis: Bishop volume comparison, Cheeger-Gromoll splitting theorem, diameter/injectivity radius estimates and eigenvalue estimates etc. If time permitted, we also try to cover some gradient estimates.
3.Immersion of submanifold, 2nd fundmental form and shape operators. Deforming hyper-surface in the direction of mean curvature flow; work of G. Huisken on covex hypersurfaces and some recent progress on mean curvature flow.
4.Ricci flow: the fundmental work of R. Hamilton, for example his work on three manifold with positive ricci curvature, his work on riemannian surface as well as in four dimension will be introduced. If time permitted, we also talk about kahler ricci flow, which may include Chen-Tian’s theorem and some other results on Kahler Ricci flow. Recent progress on Ricci flow will be discussed
Author: Xiuxiong Chen