Fredholm Eigenvalues and Quasiconformal Geometry of Polygons

On the Extremality of Harmonic Beltrami Coefficients

“…The Teichmüller and Grunsky norms are intrinsically connected with quasiconformal reflections, Fredholm eigenvalues and other quasiinvariants of quasiconformal curves. We outline briefly the main notions; the details see, e.g., in [1,[14][15][16].…”

“…It still remains open and far from its complete solving. The known results in this direction are presented, for example, in [14,16,20].…”

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

“…Its proof is much more complicated. It is based on an important theorem from [17] whose proof involves the deep results on the Gaussian curvature, Grunsky inequalities and complex geometry of universal Teichmüller space. Then µ is extremal in its class and the corresponding quasiconformal automorphism f µ of C admits the equalities Consider the map f c 0 (z) = g −q • f µ (z), where c 0 = −q coincides with the value of µ on the subarc γ 0 and is the Beltrami coefficient of the map g −q (ω) (inverse to affine deformation g q ).…”

“…For arbitrary quasidisks D, the corresponding set A 2 1 (D) is characterized similar to (4), but in more complicated way (see [17]).…”

“…The Teichmüller and Grunsky norms are intrinsically connected with quasiconformal reflections, Fredholm eigenvalues and other quasiinvariants of quasiconformal curves. We outline briefly the main notions; the details see, e.g., in [2], [12], [17], [22].…”