Illinois Journal of Mathematics

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

Jeffrey Diller

Abstract

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

Article information

Source
Illinois J. Math., Volume 45, Number 2 (2001), 453-485.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138350

Digital Object Identifier
doi:10.1215/ijm/1258138350

Mathematical Reviews number (MathSciNet)
MR1878614

Zentralblatt MATH identifier
0988.30028

Citation

Diller, Jeffrey. Green's functions, electric networks, and the geometry of hyperbolic Riemann surfaces. Illinois J. Math. 45 (2001), no. 2, 453--485. doi:10.1215/ijm/1258138350. https://projecteuclid.org/euclid.ijm/1258138350