Anatoly Golberg, Holon Institute of Technology, Israel

Local conformality conditions in plane and higher dimensions

 

Abstract

There are various necessary conditions for local conformality (that is comformality of mappings at a point) in the complex plane, like preservation of angles, independence of stretching on direction, asymptotic homogeneity, circle-like behavior, quasisymmetry, etc. Each of those can be treated as a local weak conformality condition. In higher dimensions, the situation becomes rather rigid, since due to the classical Liouville theorem, conformal mappings even in R3 are reduced to the Möbius transformations, i.e. to finite compositions of reflections across the spheres. On the other hand, many classes of mappings, e.g. quasiconformal mappings, mappings of finite distortion, mappings with controlled moduli admit differentiability almost everywhere and Hölder continuity, which also can be regarded as the weak conformality conditions.

In the talk, we discuss various aspects of local weak conformality, their relations and provide the corresponding sufficient conditions. All these results can be regarded as extensions of the classical Teichmüller-Wittich-Belinskiǐ theorem.