Boundary regularity of mass-minimizing integral currents and a question of Almgren

TL;DR

This paper announces a new boundary regularity result for area-minimizing currents in higher codimension, answering a question by Almgren and highlighting differences between boundary and interior regularity.

Contribution

It provides the first general boundary regularity theorem for area-minimizing currents without strong boundary assumptions, extending Almgren's work.

Findings

Boundary regularity is weaker than interior regularity.

Answer to Almgren's boundary regularity question.

Elementary boundary monotonicity formulae derived.

Abstract

This short note is the announcement of a forthcoming work in which we prove a first general boundary regularity result for area-minimizing currents in higher codimension, without any geometric assumption on the boundary, except that it is an embedded submanifold of a Riemannian manifold, with a mild amount of smoothness ($C^{3, a_0}$ for a positive $a_0$ suffices). Our theorem allows to answer a question posed by Almgren at the end of his Big Regularity Paper. In this note we discuss the ideas of the proof and we also announce a theorem which shows that the boundary regularity is in general weaker that the interior regularity. Moreover we remark an interesting elementary byproduct on boundary monotonicity formulae.