Boundary Regularity for Area-Minimizing Currents

Abstract

In this thesis, we settle an old question of William Allard's thesis from 1969, on boundary regularity of area-minimizing $m$-currents in arbitrary codimension, with higher boundary multiplicity and a convexity assumption. This setting generalizes Allard's celebrated theorem from 1975 on boundary regularity of area-minimizing currents to a higher boundary multiplicity minimizing setting. We develop a regularity theory that answers Allard's 1969 question by proving the $(m-3)$-rectifiability of the singular set. We show this regularity theory to be dimensionally sharp by constructing an example of an area-minimizing current with a boundary singularity for $m=3$. The starting point of the regularity theory is the uniqueness of the tangent cone at minimum density points, which we also establish in this thesis. The question of regularity for higher multiplicity boundaries, which we address in Allard's original form, was raised in a broader framework by Brian White in the Proceedings of the 1984 AMS Summer Institute, a famous collection of open problems in Geometric Measure Theory. Part of the regularity theory is in collaboration with Reinaldo Resende. Additionally, in collaboration with Camillo de Lellis, we generalize a theorem of Besicovitch from 1956 on classical rectifiability for curves in the two-dimensional plane to arbitrary dimension of the set and of the ambient space. As an application, we simplify the setup of the Naber-Valtorta technique, an extremely powerful and flexible tool for proving the rectifiability of singular sets in geometric analysis, which has been used for a variety of different problems. Together with Reinaldo Resende, we use this theorem to simplify part of the boundary regularity theory we develop in this thesis.