Past lectures

Abstract

For any two-dimensional surface f : Σ → Rn, the Willmore functional is given by 1􏰂⃗2

Σ

where H⃗ is the mean curvature vector and μ is the area measure. The central geometric feature of the functional is its invariance under the M ̈obius group of Rn. The corresponding Euler Lagrange equation is a fourth order quasilinear elliptic system, whose solutions are called Willmore surfaces.

The lecture starts with classical material on the geometry of the functional, including basic formulae and inequalities due to Willmore and Li-Yau. The main focus will then be on an- alytic questions which have been addressed in joint work with Reiner Sch ̈atzle (Universit ̈at Tu ̈bingen) in the last years. Specifically, we plan to discuss the Willmore flow, the remov- ability of point singularities in codimension one Willmore surfaces and, if times permits, a recent bilipschitz estimate for surfaces of low Willmore energy.

The course requires no prerequisites. At a later point, we will need one or two results from the literature which will be stated without proof.