Description
Book SynopsisThe monograph is devoted to the use of the moduli method in mapping theory, in particular, the meaning of direct and inverse modulus inequalities and their possible applications. The main goal is the development of a modulus technique in the Euclidean space and some metric spaces (manifolds, surfaces, quotient spaces, etc.). Particular attention is paid to the local and boundary behavior of mappings, as well as to obtaining modulus inequalities for some classes. The reader is invited to familiarize himself with all the main achievements of the author, synthesized in this book. The results presented here are of a high scientific level, are new and have no analogues in the world with such a degree of generality.
Table of ContentsGeneral definitions and notation.- Boundary behavior of mappings with Poletsky inequality.- Removability of singularities of generalized quasiisometries.- Normal families of generalized quasiisometries.- On boundary behavior of mappings with Poletsky inequality in terms of prime ends.- Local and boundary behavior of mappings on Riemannian manifolds.- Local and boundary behavior of maps in metric spaces.- On Sokhotski-Casorati-Weierstrass theorem on metric spaces.- On boundary extension of mappings in metric spaces in the terms of prime ends.- On the openness and discreteness of mappings with the inverse Poletsky inequality.- Equicontinuity and isolated singularities of mappings with the inverse Poletsky inequality.- Equicontinuity of families of mappings with the inverse Poletsky inequality in terms of prime ends.- Logarithmic H¨older continuous mappings and Beltrami equation.- On logarithmic H¨older continuity of mappings on the boundary.- The Poletsky and V¨ais¨al¨a inequalities for the mappings with (p;q)-distortion.- An analog of the V¨ais¨al¨a inequality for surfaces.- Modular inequalities on Riemannian surfaces.- On the local and boundary behavior of mappings of factor spaces.- References.- Index.
