Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Electrostatic skeletons and condition of strict descent

Linhang Huang

Abstract

Given a precompact domain $Ω\subseteq\mathbb{R}^2$, the electrostatic skeleton of $Ω$ is defined as a positive measure inside $Ω$, supported on a set with no simple loops, which generates $\partial Ω$ 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.

Electrostatic skeletons and condition of strict descent

Abstract

Given a precompact domain , the electrostatic skeleton of is defined as a positive measure inside , supported on a set with no simple loops, which generates 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.

Paper Structure

This paper contains 18 sections, 22 theorems, 58 equations, 22 figures.

Table of Contents

  1. Introduction and Results
  2. Introduction
  3. Geometric Results
  4. Structural Results
  5. Preliminary
  6. Logarithmic Potential and Green's Function
  7. Geometry of the zero sets
  8. Electrostatic Skeleton for Quadrilaterals
  9. Proof of Theorem \ref{['theo_kite']}
  10. Isosceles Trapezoids
  11. Strict descent condition
  12. Motivating example: generic quadrilateral
  13. Parametrization of the zero sets
  14. Skeleton building algorithm
  15. Regular loops
  16. ...and 3 more sections

Key Result

Theorem 1.1

Each convex kite shape admits an electrostatic skeleton. $\blacktriangleleft$$\blacktriangleleft$

Figures (22)

  • Figure 1: Electrostatic skeletons for a triangle and the regular pentagon, with level sets of the logarithmic potentials in blue.
  • Figure 2: Electrostatic skeletons for a kite shape and an isosceles trapezoid, with level sets of the logarithmic potentials in blue
  • Figure 3: Level sets of $g_1$ and $g_3$ are highlighted in red and blue respectively. The wedge $W_{1,4}$ is gray area between $\ell_1$ and $\ell_3$ that contains the polygon.
  • Figure 4: Electrostatic skeleton for a heptagon and its corresponding partition of (regular) heptagon.
  • Figure 5: The case where $\ell_i$ and $\ell_j$ are parallel
  • ...and 17 more figures

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3: Condition of Strict Descent
  • Theorem 1.4: Main Result
  • Conjecture 1.5
  • Lemma 2.1: Riesz decomposition, see Saff2010logarithmic
  • Lemma 2.2: Equipotential Lines for Convex Domains pommerenke1975univalent
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 33 more