Calculus of Variations and Geometric Measure Theory

C. De Lellis - G. De Philippis - J. Hirsch - A. Massaccesi

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

created by delellis on 21 Feb 2018
modified by massaccesi on 17 Jul 2019


Published Paper

Inserted: 21 feb 2018
Last Updated: 17 jul 2019

Journal: Matrix Annals
Year: 2017

ArXiv: 1802.07496 PDF


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.