Course Contents
In the lecture "Mean Curvature Flow" we study the most important extrinsic geometric flow, named "Mean Curvature Flow". After a (short and dense) chapter about notations and tools how to compute and differentiate objects on immersed manifolds we start with the basic definitions about the Mean Curvature flow. After that, we prove short time existence in (almost full) detail using Schauder estimates and some tools from functional analysis. In the next chapter we study the evolution equations and important consequences using different maximum principles. Then we study the long-time behavior of the flow. We example show that the flow develops singularities in finite time and that these singularities are characterized by the curvature going to infinity. We also study the behavior of curves under the flow and prove "Grayson's theorem" which is saying that a simple closed curve becomes convex under the flow in finite time. Then it shrinks to a point looking more and more like a circle.
In this lecture, we cover big parts of the book from Carlo Mantegazza "Lecture notes on Mean Curvature Flow" (Birkhaeuser).
Literature
depending on topic
Preconditions
recommended: typically Differential geometry
In the lecture "Mean Curvature Flow" we study the most important extrinsic geometric flow, named "Mean Curvature Flow". After a (short and dense) chapter about notations and tools how to compute and differentiate objects on immersed manifolds we start with the basic definitions about the Mean Curvature flow. After that, we prove short time existence in (almost full) detail using Schauder estimates and some tools from functional analysis. In the next chapter we study the evolution equations and important consequences using different maximum principles. Then we study the long-time behavior of the flow. We example show that the flow develops singularities in finite time and that these singularities are characterized by the curvature going to infinity. We also study the behavior of curves under the flow and prove "Grayson's theorem" which is saying that a simple closed curve becomes convex under the flow in finite time. Then it shrinks to a point looking more and more like a circle.
In this lecture, we cover big parts of the book from Carlo Mantegazza "Lecture notes on Mean Curvature Flow" (Birkhaeuser).
Literature
depending on topic
Preconditions
recommended: typically Differential geometry
- Lehrende: Elena Mäder-Baumdicker
- Lehrende: Sukie-Christin Vetter
Semester: ST 2023