Electrostatic skeletons and condition of strict descent

TL;DR

This paper proves Eremenko's conjecture for symmetric quadrilaterals, showing they admit unique electrostatic skeletons, and discusses conditions for their existence using conformal geometry.

Contribution

It extends the proof of electrostatic skeleton existence to symmetric quadrilaterals and explores conditions for their existence.

Findings

Confirmed Eremenko's conjecture for symmetric quadrilaterals.

Identified a natural condition implying the existence of electrostatic skeletons.

Used conformal geometry techniques in the proof.

Abstract

Given a precompact domain $\Omega \subseteq\mathbb{R}^2$, the electrostatic skeleton of $\Omega$ is defined as a positive measure inside $\Omega$, supported on a set with no simple loops, which generates $\partial \Omega$ as an equipotential curve. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. This conjecture has since been proven for triangles and regular polygons. In this paper, we will prove the conjecture for quadrilaterals with a line of symmetry using arguments from conformal geometry. We will also discuss a natural condition that implies the existence of electrostatic skeletons.