On the harmonic measure and capacity of rational lemniscates

TL;DR

This paper investigates the harmonic measure and capacity of rational lemniscates, providing new reflection principles, estimates, and a Schwarz lemma variant for their capacity, advancing understanding of their geometric properties.

Contribution

It introduces a reflection principle for harmonic measure and new capacity estimates for rational lemniscates, along with a Schwarz lemma version for holomorphic function lemniscates.

Findings

Reflection principle for harmonic measure of rational lemniscates

Capacity estimates for lemniscates and their components

A Schwarz lemma for the capacity of holomorphic function lemniscates

Abstract

We study the lemniscates of rational maps. We prove a reflection principle for the harmonic measure of rational lemniscates and we give estimates for their capacity and the capacity of their components. Also, we prove a version of Schwarz's lemma for the capacity of the lemniscates of proper holomorphic functions.