Minimality of Simons' cones in high dimensions

Venue: Seminar Room of Physics – Via Sommarive, 14 Povo - Trento

Hour: 14.00 p.m.

• Mattia Vedovato - PhD in Mathematics

Abstract:
Just after defining the notion of perimeter of a set in ℝ^n, one typically wonders if there exist sets that (locally) minimize the perimeter while being "singular" at some point. The solution to this problem, which was completely unraveled during the last century with the contributions of many authors, is quite astonishing: such a singular perimeter minimizer exists only if the dimension of the space is greater or equal than 8. Even more precisely, the Hausdorff dimension of the singular set is bounded by n-8. The easiest example of singular minimizer is the so-called Simons' cone S, defined as the set of points (x,y) in ℝ^4 ⨉ ℝ^4 such that |x|≤|y|. In this seminar, we present a very elegant and geometrically clear proof for the minimality of S, due to De Philippis and Paolini (2009) which considerably simplifies the original proof of Bombieri, de Giorgi and Giusti (1969).

The seminar corresponds to the final exam of Geometric Measure Theory, a planned course within Vedovato's first year PHD study programme

Contact person: Francesco Serra Cassano