Calculus of Variations and Geometric Measure Theory

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

On the boundary behavior of mass-minimizing integral currents

created by dephilipp on 25 Sep 2018
modified by delellis on 06 Jul 2021


Accepted Paper

Inserted: 25 sep 2018
Last Updated: 6 jul 2021

Journal: To apper in Memoirs of the American Mathematical Society
Year: 2018

ArXiv: 1809.09457 PDF


Let $\Sigma$ be a smooth Riemannian manifold, $\Gamma \subset \Sigma$ a smooth closed oriented submanifold of codimension higher than $2$ and $T$ an integral area-minimizing current in $\Sigma$ which bounds $\Gamma$. We prove that the set of regular points of $T$ at the boundary is dense in $\Gamma$. Prior to our theorem the existence of any regular point was not known, except for some special choice of $\Sigma$ and $\Gamma$. As a corollary we answer to a question of Almgren about the connectivity of minimizers.