Minicourse on Geometric flows - 19,22,23 July 2019
A series of lectures aimed at advanced undergraduate/graduate students and researchers interested in Geometry and Analysis. The focus will be on mean curvature flow and Ricci flow, and the goal is to expose the audience to a beautiful topic and describe the current state of the art in the field.
Schedule - room A11
19/7: 10:30 - 12:00 Bourni: Introduction to mean curvature flow - Lecture 1
13:00 - 14:30 Gianniotis: Introduction to Ricci flow - Lecture 1
22/7: 10:30 - 12:00 Bourni: Introduction to mean curvature flow - Lecture 2
13:00 - 14:30 Gianniotis: Introduction to Ricci flow - Lecture 2
23/7: 10:30 - 12:00 Bourni: Introduction to mean curvature flow - Lecture 3
13:00 - 14:30 Gianniotis: Introduction to Ricci flow - Lecture 3
Introduction to mean curvature flow - Theodora Bourni
Mean curvature flow is the gradient flow of the area functional; it moves the surface in the direction of steepest decrease of area. An important motivation for the study of mean curvature flow comes from its potential geometric applications. One reason for this is that it preserves several natural curvature inequalities, an observation that has led to its use as a means of proving deep classification theorems.
Lecture 1: In the first lecture we will introduce the one dimensional version of the mean curvature flow, namely the curve shortening flow. We will discuss certain basic properties of this flow, such as short time existence and the evolution equations of important geometric quantities. Moreover we will give an overview of an incredible result of Grayson that states that any simple regular closed curve contracts to a point in finite time.
Lecture 2: We will introduce mean curvature flow of hypersurfaces in Euclidean space. As in lecture one, we will briefly mention short and long time existence as well as the evolution equations of certain geometric quantities. We will discuss Huisken's famous monotonicity formula and show how it can be used to deduce interesting properties on the shape of singularities.
Lecture 3: Ancient solutions are crucial for the understanding of the singularities of mean curvature flow. Classifying them is a very difficult problem and a satisfying answer is known in only few cases. This is the case for convex ancient solutions in dimension 1. This result was first proved in the compact case by Hamilton, Daskalopoulos and Sesum. Recently in a joint work with Langford and Tinaglia, we gave a new proof and extended this result to the non-compact case. We will present the proof of this theorem.
Prerequisites: “Curves and surfaces”, “PDE"
Introduction to Ricci flow - Panagiotis Gianniotis
Lecture 1: Basics of Riemannian Geometry - examples and spaces of constant curvature - the heat equation - Ricci flow as a tool to study Riemannian manifolds.
Lecture 2: Special examples of solutions to Ricci flow - existence of solutions in general - maximum principle and preliminary singularity analysis
Lecture 3: An overview of the Ricci flow on surfaces and three dimensional manifolds with positive Ricci curvature - the general picture - open issues