On the boundary behavior of mass-minimizing integral currents

Annalisa Massaccesi; Camillo De Lellis; Guido De Philippis; Jonas Hirsch

On the boundary behavior of mass-minimizing integral currents

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arxiv 1809.09457 v3 pith:PWQGO4L3 submitted 2018-09-25 math.AP

Camillo De Lellis , Guido De Philippis , Jonas Hirsch , Annalisa Massaccesi This is my paper

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keywords gammasigmaboundaryintegralregularsmoothalmgrenanswer

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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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