Isoperimetric inequalities in graphs and surfaces

Let M be the set of metric spaces that are either graphs with bounded degree or Riemannian manifolds with bounded geometry. Kanai proved the quasi-isometric stability of several geometric properties (in particular, of isoperimetric inequalities) for the spaces in M. Kanai proves directly these results for graphs with bounded degree; in order to prove the general case, he uses a graph (an ε-net) associated to a Riemannian manifold with bounded geometry. This paper studies the stability of isoperimetric inequalities under quasi-isometries between non-exceptional Riemann surfaces (endowed with their Poincaré metrics). The present work proves the stability of the linear isoperimetric inequality for planar surfaces (genus zero surfaces) without the condition on bounded geometry. It is also shown the stability of any non-linear isoperimetric inequality.