Illinois Journal of Mathematics

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

Jeffrey Diller

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138350

Digital Object Identifier
doi:10.1215/ijm/1258138350

Mathematical Reviews number (MathSciNet)
MR1878614

Zentralblatt MATH identifier
0988.30028

Subjects
Primary: 30F15: Harmonic functions on Riemann surfaces
Secondary: 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions) 31C20: Discrete potential theory and numerical methods

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


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