Comments on the Green's function of a planar domain

TL;DR

This paper investigates various intrinsic properties of the Green's function in multiply connected planar domains, including geodesics, curvature, and critical points, using affine scaling to analyze boundary behavior.

Contribution

It introduces a method employing affine scaling to analyze boundary behavior and intrinsic properties of Green's functions in multiply connected domains.

Findings

Quantitative boundary behavior of Green's function established

Intrinsic properties like geodesics and curvature analyzed

Critical points of Green's function characterized

Abstract

We study several quantities associated to the Green's function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and $L^2$-cohomology of the capacity metric and critical points of the Green's function. The principal idea used is an affine scaling of the domain that furnishes quantitative boundary behaviour of the Green's function and related objects.