On the Conformal Energy of Quasisymmetric and Quasimöbius Mappings

2023/05/21 by Mihai Florincescu, Florincescu, Mihai, Gaven Martin +1
Materials Science · Mathematics · #30C62 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #X-ray Diffraction in Crystallography

paper · pdf · doi:10.48550/arxiv.2305.12438

openalex publication_date 2023/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article identifies the conformal energy (or mean distortion) of extremal mappings of finite distortion with a given quasisymmetric mapping of the circle as boundary data. The conformal energy of g_o:\IS→\IS is \mathcal E(g_o)=-(1)/(4π²)\iint_\mathbbS× \mathbbSlog |g_o(ζ)-g_o(η)| dζdη lt; ∞ We give explicit formulae for the conformal energy of circle homeomorphisms directly in terms of their data. As an example, if g_o: \mathbbS → \mathbbS is an η-quasi-Möbius self homeomorphism of the unit circle, then \mathcal E(g_o) ≤ \frac1π ∫^π/2₀ log η[ \cot²(t/2)] cos(t) dt This estimate is sharp. Additionally we show how a circle homeomorphism of finite conformal energy can be uniformly approximated on \IS by mappings of strictly smaller energy.

Related

• On the properness of p-conformal energy on the Teichmüller space of a Riemann surface
• Creating and Flattening Cusp Singularities by Deformations of Bi-conformal Energy
• Analytic Characterization of t-Quasicircles and Conformal Mappings onto t-Quasidisks
• A Commentary on Teichmüller's paper "Untersuchungen über konforme und quasikonforme Abbildungen" (Investigations on conformal and quasiconformal mappings) (to appear in Vol. VII of the Handbook of Teichmüller theory
• Closed surfaces with bounds on their Willmore energy