Calculus of Variations and Geometric Measure Theory
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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



Inserted: 25 sep 2018
Last Updated: 25 sep 2018

Year: 2018


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.


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