Removable singularities of mappings with inverse Poletsky inequality on Riemannian manifolds
Abstract
UDC517.5
We consider open discrete mappings of Riemannian manifolds satisfying a certain modulus inequality. We analyze the possibility of continuous extension of these mappings to an isolated point of the boundary. It is proved that these mappings admit extensions of this kind if they exclude two or more points of the connected Riemannian manifold and the majorant appearing in the modulus inequality is integrable over almost all spheres.
References
E. S. Afanasieva, V. I. Ryazanov, R. R. Salimov, On mappings in the Orlicz–Sobolev classes on Riemannian manifolds, J. Math. Sci., 181, № 1, 1–17 (2012). DOI: https://doi.org/10.1007/s10958-012-0672-z
V. Gol’dshtein, E. Sevost’yanov, A. Ukhlov, On the boundary behavior of weak $(p; q)$-quasiconformal mappings, J.~Math. Sci., 270, 420–427 (2023). DOI: https://doi.org/10.1007/s10958-023-06355-8
B. Klishchuk, R. Salimov, M. Stefanchuk, On the asymptotic behavior at innity of one mapping class, Proc. Intern. Geom. Center, 16, № 1, 50–58 (2023). DOI: https://doi.org/10.15673/tmgc.v16i1.2394
O. Martio, S. Rickman, J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 448, 1–40 (1969). DOI: https://doi.org/10.5186/aasfm.1969.448
O. Martio, S. Rickman, J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 488, 1–31 (1971). DOI: https://doi.org/10.5186/aasfm.1971.488
O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Sci.-Business Media, LLC, New York (2009).
R. Salimov, Estimation of the measure of the image of the ball, Siberian Math. J., 53, № 4, 739–747 (2012). DOI: https://doi.org/10.1134/S0037446612040155
R. Salimov, To a theory of ring $Q$-homeomorphisms with respect to a $p$-modulus, J. Math. Sci., 196, № 5, 679–692 (2014). DOI: https://doi.org/10.1007/s10958-014-1685-6
R. Salimov, B. Klishchuk, On the behavior of one class of homeomorphisms at innity, Ukr. Math. J., 74, 1617–1628 (2022). DOI: https://doi.org/10.1007/s11253-023-02158-x
S. Rickman, Quasiregular mappings, Springer-Verlag, Berlin (1993). DOI: https://doi.org/10.1007/978-3-642-78201-5
J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin etc. (1971). DOI: https://doi.org/10.1007/BFb0061216
E. A. Sevost'yanov, S. A. Skvortsov, On mappings whose inverse satisfy the Poletsky inequality, Ann. Acad. Sci. Fenn. Math., 45, 259–277 (2020). DOI: https://doi.org/10.5186/aasfm.2020.4520
E. A. Sevost'yanov, On mappings with the inverse Poletsky inequality on Riemannian manifolds, Acta Math. Hungar., 167, № 2, 576–611 (2022). DOI: https://doi.org/10.1007/s10474-022-01257-8
J. M. Lee, Riemannian manifolds: an introduction to curvature, Springer, New York (1997). DOI: https://doi.org/10.1007/0-387-22726-1_7
E. A. Sevost'yanov, A. A. Markysh, On Sokhotski–Casorati–Weierstrass theorem on metric spaces, Complex Var. and Elliptic Equat., 64, № 12, 1973–1993 (2019). DOI: https://doi.org/10.1080/17476933.2018.1557155
H. Federer, Geometric measure theory, Springer, Berlin etc. (1969).
K. Kuratowski, Topology, vol. 2, Academic Press, New York, London (1968).
С. О. Скворцов, Локальна поведінка відображень метричних просторів з розгалуженням, Укр. мат. вiсн., 17, № 4, 574–593 (2020); English translation: J. Math. Sci., 254, № 3, 425–574 (2021).
W. Hurewicz, H. Wallman, Dimension theory, Princeton Univ. Press, Princeton (1948).
C. J. Titus, G. S. Young, The extension of interiority with some applications, Trans. Amer. Math. Soc., 103, 329–340 (1962). DOI: https://doi.org/10.1090/S0002-9947-1962-0137103-6
D. Ilyutko, E. Sevost'yanov, On local properties of one class of mappings on Riemannian manifolds, J. Math. Sci., 211, № 5, 660–667 (2015). DOI: https://doi.org/10.1007/s10958-015-2622-z
E. A. Sevost'yanov, S. O. Skvortsov, O. P. Dovhopiatyi, On nonhomeomorphic mappings with the inverse Poletsky inequality, J. Math. Sci., 252, № 4, 541–557 (2021). DOI: https://doi.org/10.1007/s10958-020-05179-0
D. Ilyutko, E. Sevost'yanov, On the equicontinuity of families of inverse mappings of Riemannian manifolds, J. Math. Sci., 246, № 5, 664–670 (2020). DOI: https://doi.org/10.1007/s10958-020-04771-8
Е. А. Севостьянов, С. А. Скворцов, О локальном поведении одного класса обратных отображений, Укр. мат. вестн., 15, № 3, 399–417 (2018); English translation: J. Math. Sci., 241, № 1, 77–89 (2019).
E. A. Sevost'yanov, Isolated singularities of mappings with the inverse Poletsky inequality, Mat. Stud., 55, № 2, 132–136 (2021). DOI: https://doi.org/10.30970/ms.55.2.132-136
Copyright (c) 2024 Євген Олександрович Севостьянов, Вікторія Десятка
This work is licensed under a Creative Commons Attribution 4.0 International License.
