Area growth and Green's function of Riemann surfaces
Let \(R\) be a Riemann surface of hyperbolic type endowed with the Poincaré metric (which has constant negative curvature). For \(p\in R\) let \(A(p,t)\) denote the (Poincaré) area of \(\{q\in R\): \(\text{dist}(p,q)\leq t\}\). The authors consider the condition \[ \inf_{p\in R} A(p,t)> ce^{\alpha t} \quad\text{for}\quad t>t_ 0\tag{*} \] where \(c>0\) and \(\alpha\) are constants. If \(R\) is a plane domain they show that (*) even with \(\alpha=0\) implies that \(R\) has a Green's function. On the other hand they construct a Riemann surface of infinite genus with (*) holding for \(\alpha=0.08\) that does not have a Green's function. They ask whether this is impossible for any \(\alpha<1\); for \(\alpha=1\) it is classical that (*) implies the existence of a Green's function.
scientific article; zbMATH DE number 91592
scientific article