A surface moving so as to decrease its area most efficiently is said to evolve by mean curvature flow. Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. Mean curvature flow and its variants have many striking applications in geometry, topology, material science, image processing and general relativity.

In this course we will provide a general introduction to mean curvature flow and some of its applications. In the first half of the course we will focus on the evolution up to the first singular time, which can be described in the classical framework of smooth differential geometry and partial differential equations. A central challenge is then to extend the solutions beyond the first singular time and to analyze the structure of singularities. This is crucial for the most striking applications, and will be our focus for the second half of the course.

Prerequisites:

Some basic background in geometry is required, i.e. you should be familiar with basic concepts such as surfaces, manifolds, Riemannian metrics and the second fundamental form. Some prior knowledge of PDEs is also helpful, but not strictly required.

References:

K. Ecker: Regularity Theory for Mean Curvature Flow, Birkhauser, 2004

R. Haslhofer, B. Kleiner: Mean curvature flow of mean convex hypersurfaces, CPAM, 2016

T. Ilmanen: Elliptic regularization and partial regularity for motion by mean curvature, Memoirs of the AMS, 1994

B. White: Lecture notes on Mean Curvature flow, available at https://web.math.princeton.edu/~ochodosh/MCFnotes.pdf