Extremal properties of Sobolev's Beltrami coefficients and distortion of curvelinear functionals

TL;DR

This paper investigates extremal properties of Sobolev's Beltrami coefficients and their impact on the distortion of curvilinear functionals, advancing the understanding of quasiconformal maps and related invariants.

Contribution

It introduces a new theorem for univalent functions in quasiconformal domains and characterizes extremal Beltrami coefficients, with applications to conformal invariants.

Findings

Established a general theorem for univalent functions in quasiconformal domains.

Identified new extremal Beltrami coefficients for disk maps.

Provided applications to conformal and quasiconformal invariants.

Abstract

An important problem in applications of quasiconformal analysis and in its numerical aspect is to establish algorithms for explicit or approximate determination of the basic quasiinvariant curvelinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmuller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficients of associated quasicircles. We prove a general theorem of new type answering this question for univalent functions in arbitrary quasiconformal domains and provide its applications. The results are strengthened in the case of maps of the disk and give rise to extremal Beltrami coefficients of a new type.