Half-plane capacity and conformal radius

TL;DR

This paper establishes a relationship between the half-plane capacity of a set and its hyperbolic neighborhood's Euclidean area, linking it to the conformal radius of subdomains in the unit disc.

Contribution

It introduces a new geometric estimate connecting half-plane capacity with Euclidean area of hyperbolic neighborhoods and relates it to conformal radius.

Findings

Half-plane capacity is comparable to the Euclidean area of hyperbolic neighborhoods.

A similar estimate is proven for the conformal radius of subdomains.

A simple relation between half-plane capacity and conformal radius is established.

Abstract

In this note, we show that the half-plane capacity of a subset of the upper half-plane is comparable to a simple geometric quantity, namely the euclidean area of the hyperbolic neighborhood of radius one of this set. This is achieved by proving a similar estimate for the conformal radius of a subdomain of the unit disc, and by establishing a simple relation between these two quantities.