Fredholm eigenvalues and quasiconformal geometry of polygons

An important open problem in geometric complex analysis is to establish algorithms for the explicit determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues, and the quasireflection coefficient. This is important also for the potential theory but has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory. This survey extends our previous survey of 2005 and presents the newapproaches and recent essential progress in this field of geometric complex analysis and potential theory, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals (to which the notion of Fredholm eigenvalues also can be extended).

The Universal Teichmüller Space and Function Space

In this paper, a subspace TF02,1−s,s of the universal Teichmüller space, which is related to the analytic function space F02,1−s,s, is introduced and the holomorphy of the Bers map is shown. It is also proved that the pre-Bers map is holomorphic and the prelogarithmic derivative model T˜F02,1−s,s of TF02,1−s,s is a disconnected subset of the function space F02,1−s,s. Moreover, several equivalent descriptions of elements of TF02,1−s,s are obtained and the holomorphy of higher Bers maps is proved.