Mappings with bounded distortion and geometric structures

Authors

Boris Apanasov, University of Oklahoma, NormanFollow

Abstract

We survey recent results on connections of mappings with bounded distortion and dynamics of discrete action in different geometries. Our new tool based on quasiconformal and conformal dynamics of discrete group actions in 3-geometries at infinity of negatively curved symmetric rank one spaces is used to construct new types of quasiconformal, quasiregular and quasisymmetric mappings in space. This tool has close relations to new effects in Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds/orbifolds and non-trivial hyperbolic 4-cobordisms, to the hyperbolic and conformal interbreedings, as well as to non-faithful discrete representations of uniform hyperbolic 3-lattices.

Leaving applications of our approach to geometry and topology of manifolds to another our papers, here we discuss applications of our constructions to long standing problems for mappings with bounded distortion in different geometries, especially M.A.Lavrentiev surjectivity problem in Euclidean space, Pierre Fatou problem on radial limits and Matti Vuorinen injectivity and asymptotics problems for bounded quasiregular mappings in the unit 3-ball, as well as a possibility of such applications in spaces with complex and spherical CR-structures.

First Page

121

Last Page

146

References

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Recommended Citation

Apanasov, Boris (2022) "Mappings with bounded distortion and geometric structures," Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 5: Iss. 3, Article 2.
DOI: https://doi.org/10.56017/2181-1318.1225