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Local Hölder continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry

  • Abraham Enrique Muñoz Flores
  • Stefano NardulliEmail author
Original Paper
  • 19 Downloads

Abstract

For a complete noncompact connected Riemannian manifold with bounded geometry \(M^n\), we prove that the isoperimetric profile function \(I_{M^n}\) is a locally \((1-\frac{1}{n})\)-Hölder continuous function and so in particular it is continuous. Here for bounded geometry we mean that M have Ricci curvature bounded below and volume of balls of radius 1, uniformly bounded below with respect to its centers. We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own interest about the approximation of finite perimeter sets with finite volume by open bounded with smooth boundary ones of the same volume. Finally the upper semicontinuity of the isoperimetric profile for every metric (not necessarily complete) is shown.

Keywords

Hölder continuity of isoperimetric profile Bounded geometry Finite perimeter sets 

Mathematics Subject Classification

49Q20 58E99 53A10 49Q05 

Notes

Acknowledgements

The second author is indebted to Pierre Pansu for inspiring this paper and then to Pierre Pansu, Frank Morgan, Andrea Mondino, and Luigi Ambrosio for useful discussions on the topics of this article. The first author wish to thank the CAPES for financial support for the period in which he was a Ph.D. student at IM-UFRJ. Finally we want to thank a lot the anonymous referee whose comments contributed to improve both the results and the presentation of the proofs contained in this paper.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Abraham Enrique Muñoz Flores
    • 1
    • 2
  • Stefano Nardulli
    • 1
    Email author
  1. 1.Departamento de Matemática, Instituto de MatemáticaUFRJ-Universidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Departamento de Geometria e Representação Gráfica, Instituto de Matemática e EstatísticaUERJ-Universidade Estadual do Rio de JaneiroRio de JaneiroBrazil

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