22 jun 2015

8 hours course

Abstract.

The Willmore conjecture, proposed in 1965, concerns the quest to find the best torus of all. It is posed as a global problem in the calculus of variations: that of minimizing the Willmore energy, defined as the total integral of the square of the mean curvature, of a surface of genus one in three-space. This is a conformally invariant variational problem that has inspired a lot of mathematics over the years, helping bringing together ideas from subjects like conformal geometry, partial differential equations, algebraic geometry and geometric measure theory. In this minicourse we will present our solution to the conjecture, obtained jointly with Andre Neves, through the min-max approach. The key insight comes from the analysis of the geometric and topological properties of a new kind of sweepout: a five-dimensional family of surfaces in the three-sphere that detects the Clifford torus as a min-max minimal surface. The implementation of the program is based on the Almgren-Pitts min-max theory for the area functional, developed in the framework of Geometric Measure Theory.