On the Extremality of Harmonic Beltrami Coefficients

Abstract

We prove a general theorem, which provides a broad collection of univalent functions with equal Grunsky and Teichmüller norms and thereby the Fredholm eigenvalues and the reflection coefficients of associated quasicircles. It concerns an important problem to establish the exact or approximate values of basic quasiinvariant functionals of Jordan curves, which is crucial in applications and in the numerical aspect of quasiconformal analysis.

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

“…Recently, the author found in [13] an important application of such coefficients to geometric problems of Teichmüller space theory. Other new types of not canonical extremal Beltrami coefficients are given in [18]. The present paper continues this line, and Theorem 1 provides, in particular, that the structure of such coefficients can be rather pathological.…”

“…For such coefficients, the required smoothness of µ is provided by the representations ( 8), (9), while vanishing on γ follows from (5). A special case of Theorem 3 has been obtained in [18].…”

Quasiconformal Homeomorphisms Explicitly Determining the Basic Curve Quasi-Invariants

“…Note also that both norms κ D * and k( f ) are continuous logarithmically plurisubharmonic functions on T [28]. Now, given a function f ∈ Σ Q , take its extremal extension f µ (i.e., such that k( f ) = µ ∞ ) and set µ * = µ/ µ ∞ and pass to maps f tµ * (z) with |t| < 1.…”