Area minimizing & almost-minimizing currents: singularities and tangent cones

Seminar in Mathematics

Abstract: The Plateau problem concerns the surfaces of least m-dimensional area spanning a given (m-1)-dimensional boundary. To guarantee existence of minimizers and desirable compactness properties for sequences of surfaces, one must consider a weak notion of surface, which allows area-minimizing "surfaces” to have singularities. Two particularly natural frameworks for this problem are integral currents and mod(q) currents, which have both been studied in great depth since the 1950s, pioneered by works of De Giorgi, Federer & Fleming, Almgren, Taylor and White, and built upon by many others. The former framework allows for surfaces to have integer multiplicities, while the latter allows for multiplicities modulo a fixed integer q. I will explain the history of the problem and some recent breakthroughs in the regularity theory for each framework, as well as regularity (and failure thereof) for almost area-minimizers.

This is based on a series of joint works with Camillo De Lellis & Paul Minter, joint work with Paul Minter, Davide Parise & Luca Spolaor, and joint work with Max Goering.