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The 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.
"The book may be of interest for teachers and researchers in complex analysis and, more generally, those in function theory. Since 2009, there have been a great volume of facts derived from this theory of mappings, particularly those defined by modular inequalities. Therefore, such a book as this is very useful. The style of the author is that of a professor teaching a course to students, very clear; he tries to be as self-contained . ." (Mihai Cristea, Mathematical Reviews, February, 2025)
Evgeny Sevost'yanov is Head of Department of Mathematical Analysis, Business Analysis and Statistics at Zhytomyr Ivan Franko State University, Ukraine. His research interests include mapping theory and its applications to equations with partitional derivatives. The main results of E. Sevost'yanov concern the local and boundary behavior of mappings, mappings of metric spaces and Riemannian manifolds, quotient spaces and Riemannian surfaces. From 2002 up to now E. Sevost'yanov works as a researcher in Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine (located in Donetsk up to 2014, and in Slov'yans'k from 2014). From 2014 up to now E. Sevost'yanov works in Zhytomyr Ivan Franko State University, which belongs to the system of Ministry of Education and Science of Ukraine. Candidate of Physical and Mathematical Sciences (2006), Doctor of Physical and Mathematical Sciences (2013), Senior Researcher (2011).
General 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.
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