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The Loewner energy of loops and regularity of driving functions

Steffen Rohde, Yilin Wang

TL;DR

The Loewner energy of a rooted planar loop is introduced and the regularity result is used to show the independence of this energy from the basepoint.

Abstract

Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a $C^{1,β}$ curve (differentiable parametrization with $β$-Hölder continuous derivative) is in the class $C^{1,β-1/2}$ if $1/2<β\leq 1$, and in the class $C^{0,β+ 1/2}$ if $0 \leq β\leq 1/2$. This is the converse of a result of Carto Wong and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint.

The Loewner energy of loops and regularity of driving functions

TL;DR

The Loewner energy of a rooted planar loop is introduced and the regularity result is used to show the independence of this energy from the basepoint.

Abstract

Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a curve (differentiable parametrization with -Hölder continuous derivative) is in the class if , and in the class if . This is the converse of a result of Carto Wong and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint.

Paper Structure

This paper contains 15 sections, 32 theorems, 144 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Loop Loewner energy
  3. Chordal Loewner energy
  4. Loop Loewner energy
  5. Root-invariance for sufficiently regular loops
  6. Root-invariance for finite energy loops
  7. Proof of Theorem \ref{['thm_main']}
  8. Notations
  9. Regularity of mapped-out curves.
  10. The driving function of the initial bit of the curve.
  11. Proof of Theorem \ref{['thm_main']}
  12. Comments
  13. The sharpness of Theorem \ref{['thm_main']}
  14. Finite energy and slow spirals
  15. Consequences of Theorem \ref{['thm_main_2']}

Key Result

Theorem 1.1

The loop Loewner energy is root-invariant.

Figures (8)

  • Figure 1: Maps in the proof of Proposition \ref{['prop_quasicircle']}. Solid lines are the boundary of domains.
  • Figure 2: Illustrations of the surgeries made in the proof of Proposition \ref{['prop_finite_loop']} (left) and Propsition \ref{['prop_regular_invariant']} (right). Left: $\tilde{\gamma}$ is the loop obtained from replacing $\gamma[1/4, 1/2]$ by the hyperbolic geodesic in the complement of $\gamma[-1/2, 1/4]$. Right: $x$ and $y$ separates the solid loop into $\gamma_1$ and $\gamma_2$, $\gamma_3$ is formed by concatenation of circular arcs and replaces $\gamma_2$ in the proof.
  • Figure 3: Finite energy geodesic pairs in $\mathbb{H}$ between $0$ and $\infty$ passing through different points on the unit circle. Simulation by Brent Werness.
  • Figure 4: Illustration of different maps considered in Section \ref{['sec_regularity']}. We define the map $\Psi$ according to the value of $\beta$, and $\mu_s$ is the Möbius function defined in Corollary \ref{['cor_1.5_2']}.
  • Figure 5: $C^1$ curve $\gamma$ without bottle-necks $\leq R$.
  • ...and 3 more figures

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: wong2014 and lindtran2014regularity
  • Theorem 1.4: EE2001
  • Theorem 1.5
  • Definition 2.1: Chordal Loewner energy
  • Corollary 2.3: Two-slit Loewner energy
  • Proposition 2.4: Conformal restriction wang2016
  • Corollary 2.5
  • Proposition 2.6: Commutation relation wang2016
  • ...and 44 more