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Conformal weldings in the Loewner equation and Weil--Petersson quasislit-disks

Fei Tao, Huaying Wei, Yaosong Yang

Abstract

A simple arc $Γ= γ(0, T]$, growing into the unit disk $\mathbb D$ from its boundary, generates a driving term $ξ$ and a conformal welding $φ$ through the Loewner differential equation. When $Γ$ is the slit of a Weil--Petersson quasislit-disk $\mathbb D\setminusΓ$, the Loewner transform and its inverse $Γ\leftrightarrow ξ$ have been well understood due to Y. Wang's work. We investigate the maps $Γ\leftrightarrow φ$ in this case, giving a description of $Γ$ in terms of $φ$.

Conformal weldings in the Loewner equation and Weil--Petersson quasislit-disks

Abstract

A simple arc , growing into the unit disk from its boundary, generates a driving term and a conformal welding through the Loewner differential equation. When is the slit of a Weil--Petersson quasislit-disk , the Loewner transform and its inverse have been well understood due to Y. Wang's work. We investigate the maps in this case, giving a description of in terms of .

Paper Structure

This paper contains 5 sections, 7 theorems, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries, the main result and its motivation
  3. Proof of Theorem \ref{['main']}
  4. Proof of the sufficiency
  5. Proof of the necessity

Key Result

Proposition 1

A sense-preserving homeomorphism $h$ of $\mathbb T$ is quasisymmetric if and only if the composition operator $V_h: g \mapsto g\circ h$ gives an isomorphism of $H^{\frac{1}{2}}(\mathbb T)$.

Figures (3)

  • Figure 1: Illustration of the construction of the map $q\colon D(0,r)\to D(0,r)$
  • Figure 2: Illustration of the construction of the quasiconformal mapping $f\colon \mathbb{D}\to \mathbb{D}$
  • Figure 3: Illustration of the construction of the conformal welding $\phi\colon I^{+}\cup I^{-}\to I^{+}\cup I^{-}$

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Theorem 1: see LindSharp05MarshallRohde05
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 5 more