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Isoperimetric Inequalities in Riemann Surfaces and Graphs

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Abstract

A celebrated theorem of Kanai states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds (with positive injectivity radius) and graphs. Our main result states that we can study the (Cheeger) isoperimetric inequality in a Riemann surface by using a graph related to it, even if the surface has injectivity radius zero (this graph is inspired in Kanai’s graph, but it is different from it). We also present an application relating Gromov boundary and isoperimetric inequality.

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Authors and Affiliations

  1. Departamento de Análisis Económico y Finanzas, Universidad de Castilla-La Mancha, Facultad de CC. Sociales de Talavera, Avda. Real Fábrica de Seda, s/n 45600 Talavera de la Reina, Toledo, Spain

    Álvaro Martínez-Pérez

  2. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain

    José M. Rodríguez

Authors
  1. Álvaro Martínez-Pérez
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  2. José M. Rodríguez
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Correspondence to Álvaro Martínez-Pérez.

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Á. Martínez-Pérez: Supported in part by a grant from Ministerio de Ciencia, Innovación y Universidades (PGC2018-098321-B-I00), Spain.

J. M. Rodríguez: Supported in part by two grants from Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain.

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Martínez-Pérez, Á., Rodríguez, J.M. Isoperimetric Inequalities in Riemann Surfaces and Graphs. J Geom Anal 31, 3583–3607 (2021). https://doi.org/10.1007/s12220-020-00407-0

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