Isoperimetric inequalities in Riemann surfaces of infinite type

ds jdzj jzj With this metric S is a complete Riemannian manifold with constant curvature The only Riemann surfaces which are left out are the sphere the plane the punctured plane and the tori It is convenient to remark that this de nition of hyperbolic Rie mann surface is not universally accepted since sometimes the word hyperbolic refers to the existence of Green s function We say that S satis es the hyperbolic isoperimetric inequality HII if S is a hyperbolic Riemann surface and there exists a constant h such that for every relatively compact domain an open and connected set G with smooth boundary one has that