Martínez Pérez, ÁlvaroRodríguez, José M.2025-12-162025-12-162021Martínez-Pérez, Á., Rodríguez, J.M. Isoperimetric Inequalities in Riemann Surfaces and Graphs. J Geom Anal.2021; 31: 3583–3607.1050-69261559-002X10.1007/s12220-020-00407-0https://hdl.handle.net/20.500.14352/129071A 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.engjournal articleopen accessCheeger isoperimetric constantGromov hyperbolicityIsoperimetric inequalityPoincaré metricRiemann surfaceGeometría diferencial1204.04 Geometría Diferencial