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arXiv · 1802.07496

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

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

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.

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Camillo De Lellis, Guido De Philippis, Jonas Hirsch, Annalisa Massaccesi. 2018-02-21. Boundary regularity of mass-minimizing integral currents and a question of Almgren. https://arxiv.org/abs/1802.07496

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