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

Boundary behaviour of open, light mappings in metric measure spaces

“…Quasiregular mappings on metric measure spaces were studied in [6,9,14,21,22,43]. Finally, homeomorphisms and open, discrete mappings satisfying generalized modular inequalities were studied in [4,5,30,50,51,[57][58][59][60][61]64] on generalized metric measure spaces, other then R n with the euclidean metric. In [30] the boundary behaviour and equicontinuity of bounded open, discrete mappings on Riemannian manifolds for which a Poleckii type modular inequality holds is studied (see Theorem 5.4 in [30]).…”

Prime ends on the Riemann surfaces

“…The theory of the boundary behavior in the prime ends for the mappings with finite distortion has been developed in [12] for the plane domains and in [15] for the spatial domains. The pointwise boundary behavior of the mappings with finite distortion in regular domains on Riemann surfaces was recently studied by us in [31]. Moreover, the problem was investigated in regular domains on the Riemann manifolds for n ≥ 3 as well as in metric spaces, see e.g.…”

“…For basic definitions and notations, discussions and historic comments in the mapping theory on the Riemann surfaces, see our previous papers [30]- [32].…”

“…Here δ(E) = sup p 1 ,p 2 ∈S δ(p 1 , p 2 ) denotes the diameter of a set E in S with respect to an arbitrary metric δ in S agreed with its topology, see [30]- [31]. Correspondingly to the definition, a chain of cross-cuts σ m generates a sequence of domains d m ⊂ D such that d 1 ⊃ d 2 ⊃ .…”

“…., the domain d m contains chains of cross-cuts {σ ′ k } in all prime ends P l except their finite collection. Now, let D be a domain in the compactification S of a Riemann surface S by Kerekjarto-Stoilow, see a discussion in [30]- [31]. Open neighborhoods of points in D is induced by the topology of S. A basis of neighborhoods of a prime end P of D can be defined in the following way.…”