Quasiconformal features and Fredholm eigenvalues of convex polygons

TL;DR

This paper introduces a new approach using affine transformations to compute key functionals like Teichmuller and Grunsky norms, Fredholm eigenvalues, and quasireflection coefficients for convex polygons, a longstanding open problem in geometric complex analysis.

Contribution

It proposes a novel method based on affine transformations to explicitly determine fundamental functionals for convex polygons, advancing understanding in Teichmuller space geometry.

Findings

New algorithm for convex polygons' functionals

Explicit computation of Fredholm eigenvalues

Enhanced understanding of quasiconformal maps

Abstract

An important open problem in geometric complex analysis is to find algorithms for explicit determination of basic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmuller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of such polygons with the geometry of the universal Teichmuller space. We provide a new approach, based on affine transformations of univalent functions.