Isoperimetric Inequalities in Riemann Surfaces and Graphs Articles uri icon

publication date

  • January 2021

issue

  • 4

volume

  • 31

International Standard Serial Number (ISSN)

  • 1050-6926

Electronic International Standard Serial Number (EISSN)

  • 1559-002X

abstract

  • (copyright) 2020, Mathematica Josephina, Inc.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.

keywords

  • cheeger isoperimetric constant; gromov hyperbolicity; isoperimetric inequality; poincarĂ© metric; riemann surface