The Loewner energy of loops and regularity of driving functions

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 $C^{1,\beta }$ curve (differentiable parametrization with $\beta$-Hölder continuous derivative) is in the class $C^{1,\beta -1/2}$ if $1/2<\beta \le 1$, and in the class $C^{0,\beta +1/2}$ if $0\le \beta \le 1/2$. 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.

36 pages, 8 figures, to appear in Int. Math. Res. Not. IMRN

References in corpus (4)

• A Sharp Condition for the Loewner Equation to Generate Slits
• Reversibility of chordal SLE
• Exact Solutions for Loewner Evolutions
• Equivalent Descriptions of the Loewner Energy

Cited by in corpus (9)

• Equivalent Descriptions of the Loewner Energy
• Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines
• Large deviations of Schramm-Loewner evolutions: A survey
• A note on Loewner energy, conformal restriction and Werner's measure on self-avoiding loops
• Large deviations of multichordal SLE${}_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians
• The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles
• Large deviations of radial SLE${}_{\infty }$
• Two-loop Loewner potentials
• A deterministic approach to Loewner-energy minimizers