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


De Lellis, Camillo; De Philippis, Guido; Hirsch, Jonas; Massaccesi, Annalisa (2019). Boundary regularity of mass-minimizing integral currents and a question of almgren. In: Wood, David R; de Gier, Jan; Praeger, Cheryl E; Tao, Terence. 2017 MATRIX Annals. Cham: Springer, 193-205.

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.

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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Additional indexing

Item Type:Book Section, not_refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1 January 2019
Deposited On:28 Jun 2019 12:39
Last Modified:15 Apr 2021 15:07
Publisher:Springer
Number:2
ISBN:9783030041601
OA Status:Closed
Publisher DOI:https://doi.org/10.1007/978-3-030-04161-8_14

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