Local and boundary behavior of maps in metric spaces

Sevost′yanov, E.

doi:10.1090/spmj/1475

Local and boundary behavior of maps in metric spaces
HTML articles powered by AMS MathViewer

by E. A. Sevost′yanov
Translated by: S. Kislyakov

St. Petersburg Math. J. 28 (2017), 807-824

DOI: https://doi.org/10.1090/spmj/1475

Published electronically: October 2, 2017

PDF | Request permission

Abstract:

The paper is devoted to maps of metric spaces whose quasiconformal characteristic satisfies certain restrictions of integral nature. In particular, theorems on equicontinuity of a family of homeomorphisms are obtained, as well as some results of their extendibility to isolated points of the boundary of a domain. Under some additional topological assumptions, similar statements hold for more general open discrete maps.

References

Tomasz Adamowicz and Nageswari Shanmugalingam, Non-conformal Loewner type estimates for modulus of curve families, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 2, 609–626. MR 2731711, DOI 10.5186/aasfm.2010.3538
Cabiria Andreian Cazacu, On the length-area dilatation, Complex Var. Theory Appl. 50 (2005), no. 7-11, 765–776. MR 2155443, DOI 10.1080/02781070500087840
Christopher J. Bishop, Vladimir Ya. Gutlyanskiĭ, Olli Martio, and Matti Vuorinen, On conformal dilatation in space, Int. J. Math. Math. Sci. 22 (2003), 1397–1420. MR 1980177, DOI 10.1155/S0161171203110034
E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999
Mihai Cristea, Local homeomorphisms having local $\textrm {ACL}^n$ inverses, Complex Var. Elliptic Equ. 53 (2008), no. 1, 77–99. MR 2380822, DOI 10.1080/17476930701666924
Mihai Cristea, Open discrete mapping having local $ACL^n$ inverses, Complex Var. Elliptic Equ. 55 (2010), no. 1-3, 61–90. MR 2599611, DOI 10.1080/17476930902998985
F. W. Gehring, The definitions and exceptional sets for quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I 281 (1960), 28. MR 0124488
Anatoly Golberg, Homeomorphisms with finite mean dilatations, Complex analysis and dynamical systems II, Contemp. Math., vol. 382, Amer. Math. Soc., Providence, RI, 2005, pp. 177–186. MR 2175886, DOI 10.1090/conm/382/07057
Anatoly Golberg, Differential properties of $(\alpha ,Q)$-homeomorphisms, Further progress in analysis, World Sci. Publ., Hackensack, NJ, 2009, pp. 218–228. MR 2581624, DOI 10.1142/9789812837332_{0}015
Bent Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171–219. MR 97720, DOI 10.1007/BF02404474
Vladimir Gutlyanskii, Vladimir Ryazanov, Uri Srebro, and Eduard Yakubov, The Beltrami equation, Developments in Mathematics, vol. 26, Springer, New York, 2012. A geometric approach. MR 2917642, DOI 10.1007/978-1-4614-3191-6
Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
Juha Heinonen and Pekka Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. MR 1654771, DOI 10.1007/BF02392747
Stanislav Hencl and Pekka Koskela, Lectures on mappings of finite distortion, Lecture Notes in Mathematics, vol. 2096, Springer, Cham, 2014. MR 3184742, DOI 10.1007/978-3-319-03173-6
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
D. P. Ilyutko and E. A. Sevost′yanov, On boundary behavior of open discrete mappings on Riemannian manifolds, arXiv:1508.03278v2.
Tadeusz Iwaniec and Gaven Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001. MR 1859913
Andreĭ A. Ignat′ev and Vladimir I. Ryazanov, Finite mean oscillation in mapping theory, Ukr. Mat. Visn. 2 (2005), no. 3, 395–417, 443 (Russian, with English and Russian summaries); English transl., Ukr. Math. Bull. 2 (2005), no. 3, 403–424. MR 2325895
Andreĭ A. Ignat′ev and Vladimir I. Ryazanov, On the theory of the boundary behavior of spatial mappings, Ukr. Mat. Visn. 3 (2006), no. 2, 199–211, 301 (Russian, with English and Russian summaries); English transl., Ukr. Math. Bull. 3 (2006), no. 2, 189–201. MR 2325572
K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
Olli Martio, Vladimir Ryazanov, Uri Srebro, and Eduard Yakubov, Moduli in modern mapping theory, Springer Monographs in Mathematics, Springer, New York, 2009. MR 2466579
O. Martio, S. Rickman, and J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 448 (1969), 40. MR 0259114
O. Martio, S. Rickman, and J. Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 465 (1970), 13. MR 0267093
O. Martio, S. Rickman, and J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 488 (1971), 31. MR 299782
Raimo Näkki, Boundary behavior of quasiconformal mappings in $n$-space, Ann. Acad. Sci. Fenn. Ser. A I 484 (1970), 50. MR 0285707
Raimo Näkki and Bruce Palka, Uniform equicontinuity of quasiconformal mappings, Proc. Amer. Math. Soc. 37 (1973), 427–433. MR 0311900, DOI 10.1090/S0002-9939-1973-0311900-9
Jani Onninen and Kai Rajala, Quasiregular mappings to generalized manifolds, J. Anal. Math. 109 (2009), 33–79. MR 2585391, DOI 10.1007/s11854-009-0028-x
Kai Rajala, Mappings of finite distortion: removable singularities for locally homeomorphic mappings, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3251–3258. MR 2073299, DOI 10.1090/S0002-9939-04-07459-3
Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 994644, DOI 10.1090/mmono/073
Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
Vladimir I. Ryazanov and Ruslan R. Salimov, Weakly planar spaces and boundaries in the theory of mappings, Ukr. Mat. Visn. 4 (2007), no. 2, 199–234, 307 (Russian, with English and Russian summaries); English transl., Ukr. Math. Bull. 4 (2007), no. 2, 199–234. MR 2503889
Evgeniĭ A. Sevost′yanov, Theory of moduli and capacities, and normal families of mappings that admit branching, Ukr. Mat. Visn. 4 (2007), no. 4, 582–604, 606 (Russian, with English and Russian summaries); English transl., Ukr. Math. Bull. 4 (2007), no. 4, 573–593. MR 2513905
E. A. Sevost′yanov, On the theory of the removal of singularities for mappings with an unbounded characteristic of quasiconformality, Izv. Ross. Akad. Nauk Ser. Mat. 74 (2010), no. 1, 159–174 (Russian, with Russian summary); English transl., Izv. Math. 74 (2010), no. 1, 151–165. MR 2655240, DOI 10.1070/IM2010v074n01ABEH002483
E. A. Sevost’yanov, On the boundary behavior of open discrete mappings with unbounded characteristic, Ukrainian Math. J. 64 (2012), no. 6, 979–984. MR 3104812, DOI 10.1007/s11253-012-0693-2
E. A. Sevost′yanov and S. A. Skvortsov, On equicontinuity of generalized quasiisometries on Riemannian manifolds, arXiv:1509.02121v2.
E. S. Smolovaya, Boundary behavior of ring $Q$-homeomorphisms in metric spaces, Ukraïn. Mat. Zh. 62 (2010), no. 5, 682–689 (Russian, with Russian summary); English transl., Ukrainian Math. J. 62 (2010), no. 5, 785–793. MR 2888634, DOI 10.1007/s11253-010-0388-5
Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009
Jussi Väisälä, Modulus and capacity inequalities for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 509 (1972), 14. MR 310240
Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174, DOI 10.1007/BFb0077904