Prime ends in the Sobolev mapping theory on Riemann surfaces

Abstract

We prove criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.1. Introduction. The theory of the boundary behavior in the prime ends for the mappings with finite distortion has been developed in [11] for the plane domains and in [14] 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 [26]. Moreover, the problem was investigated in regular domains on the Riemann manifolds for n ≥ 3 as well as in metric spaces, see e. ∈ (a, b). In what follows, |γ| denotes the locus of γ, i.e. the image γ ([a, b]).We act similarly to Caratheodory ([4]) under the definition of the prime ends of domains on a Riemann surface S, see Chapter 9 in [5]. First of all, recall that a continuous mapping σ : (ii) σ m splits D into 2 domains one of which contains σ m+1 and another one σ m−1 for every m > 1;2010 Mathematics Subject Classification: 30C62.

Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces

“…The present paper is a natural continuation of our previous papers [22][23][24][25][26][27], where the reader can find the corresponding historic comments and a discussion of many definitions and relevant results. Those papers were devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemann surfaces first introduced in the plane case in paper [6] and then extended to R n , n 2, in book [7].…”

“…Hence, their compactness is equivalent to their sequential compactness, see, e.g., Remark 41.I.3 in [14], and D, D ′ , ∂D, and ∂D ′ are compact subsets of S and S ′ , correspondingly, see, e.g., Proposition I.9.3 in [2]. Thus, by Lemma 2, Remarks 1 and 2 in [26], we may assume that P 1 and P 2 are associated with the same nondegenerate component…”

“…Later on, we assume that all mappings under consideration are continuous. For the previous definitions, we refer the reader to our papers [22][23][24][25][26][27]. Here, we restrict ourselves in the main by new conceptions.…”

“…We refer the reader to [26] for definitions, notations and comments in the theory of prime ends. Now, let us start from the following statement.…”

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]).…”