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. |