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MA953 - Topics in Partial Differential Equations

Lecturer: Professor Felix Schulze

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Delivery: If possible face 2 face, otherwise time synchronous via Teams

Prerequisites: Fundamentals on the geometry of hypersurfaces are needed, as taught in MA6C0 Differential Geometry. It will be helpful to have some background on elliptic and parabolic PDE, see for example MA6A2 Advanced PDEs.

Content: Mean curvature flow is the prototype of an extrinsic curvature flow, with its intrinsic cousin being the Ricci Flow. This course will be introduction to mean curvature flow, trying to give students a direct route to current developments in the field. I will try to cover the by now classic results on curve shortening flow and convex mean curvature flow, some of the surgery techniques for mean convex mean curvature flow as well as an introduction to weak mean curvature flow and its existence theory. The course is partially based on a course I gave 2 years ago at the LSGNT, with added material. The contents below are subject to change:

  • Maximum principles for scalar functions and tensors
  • Huisken's monotonicity formula
  • Evolution of closed curves in the plane
  • Evolution of closed, convex hypersurfaces
  • Mean convex mean curvature flow and surgery
  • Brakke flows and Level Set Flow
  • A compactness theorem for integral Brakke flows
  • Existence via elliptic regularisation
  • Stratification

References:

My lecture notes for the LSGNT course can be found here. Other resources are

  • B. White, Topics in mean curvature flow, lecture notes by O. Chodosh. Available here.
  • K. Ecker, Regularity theory for Mean Curvature Flow, Birkhäuser
  • M. Ritoré and C. Sinestrari, Mean Curvature Flow and isoperimetric inequalities, Advanced Courses in Mathematics CRM Barcelona, Birkhäuser
  • R. Haslhofer, Lectures on curve shortening flow. Available here.
  • R. Haslhofer, Lectures on mean curvature flow. Available here.