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Topological regularity of isoperimetric sets in PI spaces having a deformation property

Part of: Manifolds Existence theories Global differential geometry Functions of several variables Real functions

Published online by Cambridge University Press:  09 October 2023

Gioacchino Antonelli Open the ORCID record for Gioacchino Antonelli [Opens in a new window] ,
Enrico Pasqualetto Open the ORCID record for Enrico Pasqualetto [Opens in a new window] ,
Marco Pozzetta Open the ORCID record for Marco Pozzetta [Opens in a new window]  and
Ivan Yuri Violo Open the ORCID record for Ivan Yuri Violo [Opens in a new window]
Gioacchino Antonelli
Affiliation:
Courant Institute of Mathematical Sciences, NYU, 251 Mercer Street, New York 10012, USA (ga2434@nyu.edu)
Enrico Pasqualetto
Affiliation:
Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), Jyvaskyla FI-40014, Finland (enrico.e.pasqualetto@jyu.fi)
Marco Pozzetta
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy (marco.pozzetta@unina.it)
Ivan Yuri Violo
Affiliation:
Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), Jyvaskyla FI-40014, Finland (ivan.y.violo@jyu.fi)
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Abstract

We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply, in particular, to the class of possibly collapsed $\mathrm {RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed $\mathrm {RCD}(0,N)$ spaces with Euclidean volume growth holds without the additional assumption on the boundedness of isoperimetric sets. Our strategy is of interest even in the Euclidean setting, as it simplifies some classical arguments.

Keywords

isoperimetric setPI spacedeformation propertyRCD spaceregularity

MSC classification

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 49Q20: Variational problems in a geometric measure-theoretic setting
Secondary: 26B30: Absolutely continuous functions, functions of bounded variation 26A45: Functions of bounded variation, generalizations 49J40: Variational methods including variational inequalities
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 23
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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