On the Boundary Behavior of Mappings in the Class $$W^{1,1}_\mathrm{loc}$$ on Riemann Surfaces

Afanas’eva, E.S., Ryazanov, V.I., Salimov, R.R.: On mappings in Orlicz–Sobolev classes on Riemannian manifolds. Ukr. Mat. Visn. 8(3), 319–342, 461 (2011) (in Russian). (transl. in J. Math. Sci. (N. Y.) (2012). 181(1), 1–17) Beardon, A.F.: The geometry of discrete groups. Graduate Texts in Math, vol. 91. Springer-Verlag, New York (1983) Bourbaki, N.: General topology. The main structures. Nauka, Moscow (1968). (in Russian) Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer-Verlag, New York (1969) Forster, O.: Riemannsche Flächen, Heidelberger Taschenbücher, vol. 184. Springer-Verlag, Berlin-New York (1977) Fuglede, B.: Extremal length and functional completion. Acta Math. 98, 171–219 (1957) Gutlyanskii, V., Ryazanov, V. Srebro, U., Yakubov, E.: The Beltrami equation. A geometric approach. Developments in Mathematics, vol. 26. Springer, New York (2012) Gehring, F.W., Lehto, O.: On the total differentiability of functions of a complex variable. Ann. Acad. Sci. Fenn. A1. Math. 272, 1–9 (1959) Ignat’ev, A.A., Ryazanov, V.I.: Finite mean oscillation in mapping theory. Ukr. Mat. Visn. 2(3), 395–417 (2005). (transl. in Ukrainian Math. Bull. 2005. 2(3), 403–424) Iwaniec, T., Sverak, V.: On mappings with integrable dilatation. Proc. Am. Math. Soc. 118, 181–188 (1993) Iwaniec, T., Martin, G.: Geometric function theory and non-linear analysis. Oxford Univ. Press, Oxford (2001) Kovtonyuk, D.A., Petkov, I.V., Ryazanov, V.I.: On the boundary behavior of solutions of the Beltrami equations. Ukrain. Mat. Zh. 63(8), 1078–1091 (2011). (in Russian). (transl. in Ukrainian Math. J. (2012). 63(8), 1241–1255) Kovtonyuk, D.A., Petkov, I.V., Ryazanov, V.I., Salimov, R.R.: Dirichlet problem for the Beltrami equations. Algebra i Analiz. 25(4), 101–124 (2013). (transl. in St. Petersburg Math. J (2014). 25(4), 587–603) Kovtonyuk, D., Ryazanov, V.: On the boundary behavior of generalized quasi-isometries. J. Anal. Math. 115, 103–120 (2011) Kovtonyuk, D., Ryazanov, V.: On the theory of lower Q-homeomorphisms. Ukr. Mat. Visn. 5(2), 159–184 (2008). (transl. in Ukrainian Math. Bull. 5(2), 157–181 (2008)) Krushkal’, S.L., Apanasov, B.N., Gusevskii, N.A.: Kleinian groups and uniformization in examples and problems. Transl. of Math. Mon., vol. 62. AMS, Providence, RI (1986) Kuratowski, K.: Topology, vol. 1. Academic Press, New York, London (1966) Kuratowski, K.: Topology, vol. 2. Academic Press, New York, London (1968) Lehto, O., Virtanen, K.J.: Quasiconformal mappings in the plane. Springer-Verlag, Berlin, Heidelberg (1973) Martio, O., Ryazanov, V., Srebro, U., Yakubov, E.: Moduli in Modern Mapping Theory. Springer, New York (2009) Maz’ya, V.: Sobolev Spaces. Springer-Verlag, Berlin (1985) Menchoff, D.: Sur les differentielles totales des fonctions univalentes. Math. Ann. 105, 75–85 (1931) Ryazanov, V., Salimov, R.: Weakly flat spaces and boundaries in the theory of mappings. Ukr. Mat. Visn. 4(2), 199–234,307 (2007). (transl. in Ukrainian Math. Bull. 4(2), 199–234 (2007)) Ryazanov, V., Srebro, U., Yakubov, E.: Integral conditions in the mapping theory. Ukr. Mat. Visn. 7(1), 73–87 (2010). (transl. in Math. Sci. J. 173(4), 397-407 (2011)) Ryazanov, V.I., Volkov, S.V.: On the boundary behavior of mappings in the class \(W^{1,1}_{\rm loc}\) on Riemann surfaces. In: Proceedings of Inst.Appl.Math.Mech. of NASU, vol. 29, pp. 34–53. (2015) (in Russian) Ryazanov, V., Volkov, S.: On Sobolev’s mappings on Riemann surfaces. ArXiv: 1604.00280v5 [math.CV] 15 Oct 2016, p. 27 Smolovaya, E.S.: Boundary behavior of ring Q-homeomorphisms in metric spaces. Ukrain. Mat. Zh. 62(5), 682–689 (2010). (Russian). (transl. in Ukrainian Math. J. 62(5), 785–793 (2010)) Stoilow, S.: Lecons sur les principes topologiques de la theorie des fonctions analytiques. Gauthier-Villars, Paris (1956) Uryson, P.S.: Zum Metrisationsproblem. Math. Ann. 94, 309–315 (1925) van der Waerden, B.L.: Algebra (Russian). Transl. from the German, Nauka, Moscow (1976) Zieschang, H., Vogt, E., Coldewey, H.-D.: Surfaces and planar discontinuous groups. Lecture Notes in Mathematics, vol. 835. Springer, Berlin (1980)