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Tr. Mat. Inst. Steklova, 2017, Volume 298, Pages 139–143 (Mi tm3812)  

On the Isotopy Problem for Quasiconformal Mappings

V. A. Zorich

Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia

Abstract: The question of the isotopy of a quasiconformal mapping and its special aspects in dimension greater than $2$ are considered. It is shown that an arbitrary quasiconformal mapping of a ball has an isotopy to the identity map such that the coefficient of quasiconformality (dilatation) of the mapping varies continuously and monotonically. In contrast to the planar case, in dimension higher than $2$ such an isotopy is not possible in an arbitrary domain. Examples showing specific features of the multidimensional case are given. In particular, they show that even when such an isotopy exists, it is not always possible to perform an isotopy so that the coefficient of quasiconformality approaches $1$ monotonically at each point in the source domain.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00592-а
Ministry of Education and Science of the Russian Federation НШ-9110.2016.1
This work was supported by the Russian Foundation for Basic Research (project no. 17-01-00592-a) and by a grant of the President of the Russian Federation (project no. NSh-9110.2016.1).


DOI: https://doi.org/10.1134/S0371968517030104

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English version:
Proceedings of the Steklov Institute of Mathematics, 2017, 298, 129–132

Bibliographic databases:

UDC: 517.54+514.774
Received: December 15, 2016

Citation: V. A. Zorich, “On the Isotopy Problem for Quasiconformal Mappings”, Complex analysis and its applications, Collected papers. On the occasion of the centenary of the birth of Boris Vladimirovich Shabat, 85th anniversary of the birth of Anatoliy Georgievich Vitushkin, and 85th anniversary of the birth of Andrei Aleksandrovich Gonchar, Tr. Mat. Inst. Steklova, 298, MAIK Nauka/Interperiodica, Moscow, 2017, 139–143; Proc. Steklov Inst. Math., 298 (2017), 129–132

Citation in format AMSBIB
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\vol 298
\pages 139--143
\publ MAIK Nauka/Interperiodica
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