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Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy

Jinwoo Sung, Yilin Wang

Abstract

We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformations with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of the driving function of the curve. This result gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo, which has the same variational formula. We also deduce the variation of the total mass of SLE$_{8/3}$ loops touching the Jordan curve under quasiconformal deformations.

Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy

Abstract

We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformations with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of the driving function of the curve. This result gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo, which has the same variational formula. We also deduce the variation of the total mass of SLE loops touching the Jordan curve under quasiconformal deformations.
Paper Structure (10 sections, 16 theorems, 89 equations, 2 figures)

This paper contains 10 sections, 16 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Deformation of chords in the half-plane
  3. Variation of the chordal Loewner driving function
  4. Variation of chordal Loewner energy
  5. Deformation of Weil--Petersson quasicircles
  6. Loop driving function
  7. Variation of Loewner energy for a part of the quasicircle
  8. Variation of the loop Loewner energy
  9. Remarks and open questions
  10. Large time behavior of finite-energy Loewner chains

Key Result

Theorem 1.1

Let $\eta$ be a simple chord in $(\mathbb{H}; 0, \infty)$ under capacity parametrization and $\nu \in L^\infty (\mathbb{H})$ be an infinitesimal Beltrami differential whose support is compact and disjoint from $\eta$. For $\varepsilon \in \mathbb{R}$ such that $\lVert \varepsilon\nu \rVert_{\infty} and where $\mathrm{d}^2 z$ is the Euclidean area measure, $\lambda_\cdot$ is the driving function

Figures (2)

  • Figure 1: A commutative diagram illustrating the quasiconformal maps and related Loewner chains in Section \ref{['sec:chordal-variation']}. The gray shaded areas denote the support of the Beltrami differentials. The arrows in red are quasiconformal maps, and those in black are conformal maps.
  • Figure 2: A commutative diagram illustrating the quasiconformal maps and related conformal mapping-out functions in Section \ref{['sec:WP']}. The gray shaded areas denote the support of the Beltrami differentials. The arrows in red are quasiconformal maps, and those in black are conformal maps.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2: See W2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['thm:intro_driving_function']}
  • ...and 27 more