On uniform equicontinuity of generalized quasiisometries
on Riemannian manifolds

E. A. Sevost’yanov; S. A. Skvortsov

doi:10.15330/ms.45.2.159-169

The present paper is devoted to the study of mappings with finite distortion on Riemannian manifolds. Theorems on local behavior of generalized quasiisometries with unbounded characteristic of quasiconformality are obtained. In particular, we have proved that a family of mappings $f\colon D\rightarrow {\mathbb M}_*^n$ between Riemannian manifolds ${\mathbb M}^n$ and ${\mathbb M}_*^n$ is equicontinuous whenever $f$ lies in a ball $B_R,$ $f$ does not take values from a fixed continuum $K\subset B_R,$ and a quasiconformality coefficient $Q(x)$ has a finite mean oscillation at every point.

1. D.P. Ilyutko, E.A. Sevostyanov, On local properties of one class of mappings on Riemannian manifolds, Ukr. Mat. Visnyk, 12 (2015), №2, 210-221.

2. F. Gehring, Lipschitz mappings and p-capacity of rings in n-space, Ann. of Math. Stud., 66 (1971), 175-193.

3. J.M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, New York, 1997.

4. E.G. Poznyak, E.V. Shikin, Differential Geometry: First Introduction, Moscow State University, Moscow, 1990.

5. J. Vaisala, Lectures on n.Dimensional Quasiconformal Mappings, Lecture Notes in Math., V.229, Springer.Verlag, Berlin etc., 1971.

6. V. Ryazanov, R. Salimov, Weakly flat spaces and bondaries in the mapping theory, Ukr. Math. Bull., 4 (2007), №2, 199-233.

7. B. Fuglede, Extremal length and functional completion, Acta Math, 98 (1957), 171-219.

8. R.R. Salimov, E.A. Sevostyanov, The theory of shell-based Q-mappings in geometric function theory, Sb. Math., 201 (2010), №5-6, 909-934.

9. O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in Modern Mapping Theory, Springer Monographs in Mathematics, Springer, New York etc., 2009.

10. J. Heinonen, Lectures on Analysis on metric spaces, Springer Science+Business Media, New York, 2001.

11. E.S. Afanaseva, V.I. Ryazanov, R.R. Salimov, On mappings in Orlicz-Sobolev classes on Riemannian manifolds, J. Math. Sci., 181 (2012), №1, 1.17.

12. S. Rickman, Quasiregular mappings, Results in Mathematic and Related Areas (3), 26, Springer-Verlag, Berlin, 1993.

13. T. Adamowicz, N. Shanmugalingam, Non-conformal Loewner type estimates for modulus of curve families, Ann. Acad. Sci. Fenn. Math., 35 (2010), 609-626.

14. K. Kuratowski, Topology, V.2, Academic Press, New York.London, 1968.

15. E.S. Smolovaya, Boundary behavior of ring Q-homeomorphisms in metric spaces, Ukrainian Math. J., 62 (2010), №5, 785-793.

16. S. Saks, Theory of the Integral, Dover Publ. Inc., New York, 1964.

	On uniform equicontinuity of generalized quasiisometries on Riemannian manifolds
Author	E. A. Sevost’yanov, S. A. Skvortsov esevostyanov2009@mail.ru National Academy of Sciences of Ukraine
Abstract	The present paper is devoted to the study of mappings with finite distortion on Riemannian manifolds. Theorems on local behavior of generalized quasiisometries with unbounded characteristic of quasiconformality are obtained. In particular, we have proved that a family of mappings $f\colon D\rightarrow {\mathbb M}_^n$ between Riemannian manifolds ${\mathbb M}^n$ and ${\mathbb M}_^n$ is equicontinuous whenever $f$ lies in a ball $B_R,$ $f$ does not take values from a fixed continuum $K\subset B_R,$ and a quasiconformality coefficient $Q(x)$ has a finite mean oscillation at every point.
Keywords	Riemannian manifolds; quasiconformal mappings; mappings with bounded and finite distortion; quasiisometries; moduli of families of curves
DOI	doi:10.15330/ms.45.2.159-169
Reference	1. D.P. Ilyutko, E.A. Sevostyanov, On local properties of one class of mappings on Riemannian manifolds, Ukr. Mat. Visnyk, 12 (2015), №2, 210-221. 2. F. Gehring, Lipschitz mappings and p-capacity of rings in n-space, Ann. of Math. Stud., 66 (1971), 175-193. 3. J.M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, New York, 1997. 4. E.G. Poznyak, E.V. Shikin, Differential Geometry: First Introduction, Moscow State University, Moscow, 1990. 5. J. Vaisala, Lectures on n.Dimensional Quasiconformal Mappings, Lecture Notes in Math., V.229, Springer.Verlag, Berlin etc., 1971. 6. V. Ryazanov, R. Salimov, Weakly flat spaces and bondaries in the mapping theory, Ukr. Math. Bull., 4 (2007), №2, 199-233. 7. B. Fuglede, Extremal length and functional completion, Acta Math, 98 (1957), 171-219. 8. R.R. Salimov, E.A. Sevostyanov, The theory of shell-based Q-mappings in geometric function theory, Sb. Math., 201 (2010), №5-6, 909-934. 9. O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in Modern Mapping Theory, Springer Monographs in Mathematics, Springer, New York etc., 2009. 10. J. Heinonen, Lectures on Analysis on metric spaces, Springer Science+Business Media, New York, 2001. 11. E.S. Afanaseva, V.I. Ryazanov, R.R. Salimov, On mappings in Orlicz-Sobolev classes on Riemannian manifolds, J. Math. Sci., 181 (2012), №1, 1.17. 12. S. Rickman, Quasiregular mappings, Results in Mathematic and Related Areas (3), 26, Springer-Verlag, Berlin, 1993. 13. T. Adamowicz, N. Shanmugalingam, Non-conformal Loewner type estimates for modulus of curve families, Ann. Acad. Sci. Fenn. Math., 35 (2010), 609-626. 14. K. Kuratowski, Topology, V.2, Academic Press, New York.London, 1968. 15. E.S. Smolovaya, Boundary behavior of ring Q-homeomorphisms in metric spaces, Ukrainian Math. J., 62 (2010), №5, 785-793. 16. S. Saks, Theory of the Integral, Dover Publ. Inc., New York, 1964.
Pages	159-169
Volume	45
Issue	2
Year	2016
Journal	Matematychni Studii
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On uniform equicontinuity of generalized quasiisometries on Riemannian manifolds