Independent Studies in Geometric Analysis — GEA

Instructor: Dr. Balázs CSIKÓS

Text:  C. Mantegazza, Lecture Notes on Mean Curvature Flow

Prerequisite: Calculus on smooth manifolds, basic notions of Riemannian geometry.

Description: This course focuses on the study of the mean curvature flow. The mean curvature flow is an important example of a geometric flow, which evolves a hypersurface of a Riemannian manifold with velocity equal to the mean curvature vector field. In most cases, the mean curvature flow develops singularities. The goal of the course is to understand the most important techniques to describe how the shape of a hypersurface changes under the action of the flow, how singularities are formed, and how the hypersurface behaves close to a singularity.
The mean curvature flow has several applications in geometry. There are also many analogies between the mean curvature flow and other geometric flows, e.g., the Ricci flow, which was used by Perelman in his celebrated proof of Thurston's geometrization conjecture. Thus, it is a good starting point to study geometric flows.

Topics:

• Geometry of submanifolds of a Riemannian manifold: Fundamental forms, shape operator, Gauss and Codazzi-Mainardi equations.
• First variation of the volume of a submanifold. Mean curvature vector field. Mean curvature flow. Examples. Short time existence.
• Evolution of geometric quantities: Maximum principle. Comparison principle. Evolution equations and their consequences. Convexity invariance.
• Monotonicity formula and type I singularities: Monotonicity formula for the mean curvature flow. Type I singularities and the rescaling procedure. Analysis of singularities.
• Type II singularities: Hamilton's blow up. Hypersurfaces with nonnegative mean curvature. Planar curves. Hamilton's Harnack estimate.