Strong submultiplicativity of the Poincare metric

TL;DR

This paper provides a direct proof of the strong submultiplicativity of the Poincaré metric, introduces a new Poincaré capacity for compact sets, and extends related inequalities via universal covering maps.

Contribution

It offers a direct proof of Solynin's result, defines Poincaré capacity for compact sets, and generalizes submultiplicativity inequalities using universal covering maps.

Findings

Poincaré metric is strongly submultiplicative.

Poincaré capacity coincides with logarithmic capacity for connected sets.

Extension of submultiplicativity inequalities to universal covering maps.

Abstract

We give a direct proof of an important result of Solynin which says that the Poincar\'e metric is a strongly submultiplicative domain function. This result is then used to define a new capacity for compact subsets of the complex plane $\mathbb{C}$, which might be called Poincar\'e capacity. If the compact set $K \subseteq \mathbb{C}$ is connected, then the Poincar\'e capacity of $K$ is the same as the logarithmic capacity of $K$. In this special case, the submultiplicativity is well--known and can be stated as an inequality for the normalized conformal map onto the complement of $K$. Using the connection between Poincar\'e metrics and universal covering maps this inequality is extended to the much wider class of universal covering maps.