Three Applications of Conformal Geometry Techniques to Planar Geometry Problems

Abstract

We talk about how to use conformal geometry techniques to solve problems in three different areas. For the first application, we introduce a Collet-Eckmann type condition for the unicritical laminations on the unit circle. We prove that this condition implies the lamination admits a Hölder continuous conformal welding which produces a Julia set for some unicritical polynomial. For the second application, We consider the solution of $-\Delta u = 1$ on convex domains $\Omega \subset \mathbb{R}^2$ subject to Dirichlet boundary conditions $u =0$ on $\partial \Omega$. We consider the two shape optimization problems $\| \nabla u\|_{L^{\infty}}/ |\Omega|^{1/2}$ and $\| \nabla u\|_{L^{\infty}}/ \mathcal{H}^1( \partial \Omega)$. We prove that (1) either the extremal domain does not have a $C^{2 + \varepsilon}$ boundary or (2) there exists an infinite set of points on $\partial \Omega$ where the curvature vanishes. Finally, we present a result about electrostatic skeleton. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. 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.