Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines

Fredrik Viklund, Yilin Wang

Abstract

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil-Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian free field.

Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines

Abstract

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil-Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian free field.

Paper Structure

This paper contains 15 sections, 36 theorems, 120 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Cutting and welding
  3. Flow-line identity
  4. SLE/GFF discussion and heuristics
  5. Preliminaries
  6. $H^{1/2}$ on chord-arc curves
  7. Conformal welding
  8. Loewner energy
  9. Proofs of main results
  10. Welding identity: half-plane version
  11. Welding identity: disk version
  12. Flow-line identity
  13. Complex-valued function identity
  14. Trace operators on chord-arc curves
  15. Density of $C^{\infty}_c (\mathbb{C})$ in $\mathcal{E} (\mathbb{C})$

Key Result

Theorem 1.1

Suppose $\eta$ is a Jordan curve through $\infty$, let $f$ and $g$ be conformal maps associated to $\eta$ as above, and suppose $\varphi \in \mathcal{E}(\mathbb{C})$ is given. Then we have the identity: where

Figures (3)

  • Figure 1.1: Isometric conformal welding: $h = g^{-1} \circ f$ is constructed from the measures $e^u dx$ and $e^v dx$, and their pushforward measures by $f$ and $g$ both give $e^{\varphi} |dz|$ on $\eta$.
  • Figure 3.1: Arclength isometric welding of $H_1$ and $H_2^*$. The isometry $\psi = G^{-1} \circ F|_{\eta_1}$ identifies arcs of the same length and $F, G$ are conformal maps. The welding curve $\eta$ has energy bounded by the sum of the energies of $\eta_1$ and $\eta_2$.
  • Figure 3.2: Comparing the Loewner energy of equipotentials.

Theorems & Definitions (63)

  • Theorem 1.1: Cutting
  • Theorem 1.2: Isometric conformal welding
  • Corollary 1.3
  • Remark
  • Theorem 1.4: Flow-line identity
  • Corollary 1.5
  • Remark
  • Corollary 1.6: Complex identity
  • Lemma 2.1: Adams
  • Lemma 2.2
  • ...and 53 more