Green's functions, electric networks, and the geometry of hyperbolic Riemann surfaces

Abstract

We compare Green's function $g$ on an infinite volume, hyperbolic Riemann surface $X$ with an analogous discrete function $g_{\text{disc}}$ on a graphical caricature $\mathrm{\Gamma }$ of $X$. The main result, modulo technical hypotheses, is that $g$ and $g_{\text{disc}}$ differ by at most an additive constant $C$ which depends only on the Euler characteristic of $X$. In particular, the estimate of $g$ by $g_{\text{disc}}$ remains uniform as the geometry (i.e., the conformal structure) of $X$ varies.