Mean curvature flow and ancient solutions
In this course we plan to introduce students to the mean curvature flow. After introducing Huisken's monotonicity formula, maximal time of existence result and basics of the singularity analysis we plan to derive some results for the mean convex mean curvature flow such as: the convexity and the cylindrical estimate, the gradient estimate and introduce the mean curvature flow with surgeries. We plan to mention recent results about the classification of ancient solutions in the mean curvature flow which was used to prove the Mean convex neighborhood conjecture, which says that if a singularity is cylindrical, then there is a space time neighborhood around a singularity in which the mean curvature is strictly positive.
We plan to mention recent results about the classification of ancient solutions in the mean curvature flow which was used to prove the Mean convex neighborhood conjecture, which says that if a singularity is cylindrical, then there is a space time neighborhood around a singularity in which the mean curvature is strictly positive.
Different papers that will be shared with students throughout the semester. For the basics it is ' Regularity theory for Mean Curvature Flow' by Klaus Ecker.
Partial differential equations and very basics of differential geometry (though I plan to introduce differential geometry concepts that we will use in the class).