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arXiv · 1809.09457

On the boundary behavior of mass-minimizing integral currents

Camillo De Lellis·Guido De Philippis·Jonas Hirsch·Annalisa Massaccesi
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Abstract

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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Camillo De Lellis, Guido De Philippis, Jonas Hirsch, Annalisa Massaccesi. 2018-09-25. On the boundary behavior of mass-minimizing integral currents. https://arxiv.org/abs/1809.09457

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