Quasi-symmetric mappings and their generalizations on Riemannian manifolds

Elena Afanas'eva; Viktoriia Bilet

doi:10.37069/1810-3200-2021-18-2-1

Quasi-symmetric mappings and their generalizations on Riemannian manifolds

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2021-18-2-1 ◽

2021 ◽

Vol 18 (2) ◽

pp. 145-159

Author(s):

Elena Afanas'eva ◽

Viktoriia Bilet

Keyword(s):

Riemannian Manifolds ◽

Boundary Behavior ◽

Quasiconformal Mappings ◽

Uniform Domains

A relation between $\eta$-quasi-symmetric homomorphisms and $K$-quasiconformal mappings on $n$-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of $\eta$-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian mani\-folds, which satisfy the obtained results, were also formulated. In addition, quasiballs, $c$-locally connected domains, and the corresponding results were also considered.

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On a connection between some classes of mapping on Riemannian manifolds

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2020-34-1 ◽

2021 ◽

Vol 34 ◽

pp. 3-10

Author(s):

Olena Afanas'eva ◽

Viktoriia Bilet

Keyword(s):

Riemannian Manifolds ◽

Boundary Behavior ◽

Partition Of Unity ◽

Partition Of Unity Method ◽

Finsler Manifolds ◽

Locally Finite ◽

Uniform Domains

In this paper we study the connection between $\eta$-quasisymmetric homomorphisms and $K$-quasi\-con\-for\-mal mappings on $n$-dimensional smooth connected Riemannian manifolds. The main result of our research is the Theorem 3.1. For its proof we use a partition of unity method, which subordinate to the locally finite atlas of the manifold. Several results on the boundary behavior of $\eta$-quasisymmetric homomorphisms between two arbitrary domains, QED (uniform) domains and domains with weakly flat boundaries and compact closures on the Riemannian manifolds are also obtained in view of the above relations. The obtained results can be applied to Finsler manifolds with the addition of some conditions, which will take into account the specific of the initial manifold.

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On the boundary behavior of quasiconformal mappings

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2019-16-2-8 ◽

2019 ◽

Vol 16 (2) ◽

pp. 289-300

Author(s):

Vladimir Zorich

Keyword(s):

Boundary Behavior ◽

Quasiconformal Mappings ◽

Open Questions

We discuss some open questions of the theory of quasiconformal mappings related to the field of studies of Professor G. D. Suvorov. The present work is dedicated to his memory.

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On nonhomeomorphic mappings with the inverse Poletsky inequality

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-3-7 ◽

2020 ◽

Vol 17 (3) ◽

pp. 414-436

Author(s):

Evgeny Sevost'yanov ◽

Serhii Skvortsov ◽

Oleksandr Dovhopiatyi

Keyword(s):

Boundary Behavior ◽

Hölder Continuity ◽

Continuous Extension ◽

Quasiconformal Mappings ◽

Holder Continuity ◽

Hölder Continuous ◽

Finite Distortion ◽

Mappings Of Finite Distortion ◽

Lipschitz Continuous ◽

Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

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