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arxiv: 2403.11023 · v2 · submitted 2024-03-16 · 🧮 math.CV

Carath\'eodory boundary extensions for generalized quasiregular mappings

Victoria Desyatka , Evgeny Sevost'yanov This is my paper

Pith reviewed 2026-05-24 03:39 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C65
keywords boundary extensioninverse Poletsky inequalityquasiregular mappingsfinite distortionopen discrete mappingsCarathéodory theoremmodulus condition
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The pith

Open discrete mappings obeying an inverse Poletsky inequality with integrable majorant admit continuous boundary extensions when the boundary image is finitely connected and its preimage is nowhere dense.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that mappings of domains in Euclidean space satisfying the inverse Poletsky inequality with an integrable majorant, and that are open and discrete, possess continuous extensions to the boundary of the domain. This holds whenever the image of the original boundary is finitely connected relative to the target domain, the preimage of the target boundary is nowhere dense, and suitable geometric conditions on the domains are met. The integrability condition on the majorant can be weakened to hold on almost all concentric spheres around each point. The result applies in particular to homeomorphisms and to open discrete closed mappings that obey the appropriate modulus condition. A reader would care because such extensions determine how the mappings behave at domain boundaries and therefore control their global properties in geometric function theory.

Core claim

Assume that the image of the boundary of the original domain is finitely connected relative to the mapped domain, and the preimage of the boundary of the latter is nowhere dense. Then, under certain conditions on the geometry of these domains, mappings that satisfy the inverse Poletsky inequality with an integrable majorant, and that are open and discrete, have a continuous boundary extension. The result remains valid when the majorant is integrable over almost all concentric spheres centered at each point, and it covers homeomorphisms together with open discrete closed mappings satisfying the corresponding modulus condition.

What carries the argument

The inverse Poletsky inequality with integrable majorant, together with the finite-connectivity and nowhere-dense boundary conditions, which together force the continuous boundary extension for open discrete mappings.

If this is right

  • The mappings extend continuously to the closure of the domain.
  • The extension property holds for homeomorphisms obeying the modulus condition.
  • The result covers open discrete closed mappings under the stated geometric assumptions.
  • The integrability requirement can be relaxed to almost every concentric sphere without losing the extension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The extension result may allow construction of solutions to boundary-value problems for these mappings on domains with controlled boundary geometry.
  • It may connect to modulus estimates that control distortion near boundaries in higher-dimensional settings.
  • One could test the necessity of finite connectivity by examining annular domains whose boundary images become infinitely connected.

Load-bearing premise

The image of the boundary of the original domain must be finitely connected relative to the mapped domain while the preimage of the target boundary remains nowhere dense.

What would settle it

A concrete open discrete mapping satisfying the inverse Poletsky inequality with integrable majorant on a pair of domains where the boundary image fails to be finitely connected, yet the mapping has no continuous extension to the boundary.

Figures

Figures reproduced from arXiv: 2403.11023 by Evgeny Sevost'yanov, Victoria Desyatka.

Figure 1
Figure 1. Figure 1: To the proof of Theorem 1.1 yi , i = 1, 2, . . . , may be chosen such that xi , yi 6∈ f −1 (C(f, ∂D) ∩ D ′ ). Indeed, since under condition 4) the set f −1 (C(f, ∂D) ∩ D ′ ) is nowhere dense in D, there exists a sequence xki ∈ D \ (f −1 (C(f, ∂D) ∩ D ′ )), k = 1, 2, . . . , such that xki → xi as k → ∞. Put ε > 0. Due to the continuity of the mapping f at the point xi , for the number i ∈ N there is a numbe… view at source ↗
Figure 2
Figure 2. Figure 2: An open quasiregular mapping that satisfies condit [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: ). We choose ε > 0 so that ϕ(c) 6∈ Pε, dist (Pε, ∂D ∗ ) > ε. Due to [Ku, Theorem 1.I, d 1 g a b ñ * 2 1* 2 oajg - = D D * y1 y2 j( )b j( )c j( ) a j( ) d Pe * a2 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Formulation of Lemma 5.2 Proof. By the condition of the lemma, for any z0 ∈ E and for any neighborhood U of the point z0 there is a neighborhood V ⊂ U such that (V \ E) ∩ D ′ consists of a finite number of components. Since xk → z1 as k → ∞, there are infinitely many elements of the sequence xk, k = 1, 2, . . . , that belong only to one component of (V \ E) ∩ D ′ . Moreover, all these elements belong to on… view at source ↗
Figure 5
Figure 5. Figure 5: The formulation of Lemma 5.3 Proof. The existence of paths α and β, satisfying conditions a)–c), follows by Lemma 5.2. It remains only to check the condition d), namely, to explain why the paths α and β may be chosen disjoint. Two options are possible: 1) K1 6= K2, i.e., paths α and β (except of their right endpoints) belong to (disjoint) components of D ′ \ E. Then α and β are disjoin by the definition. 2… view at source ↗
Figure 6
Figure 6. Figure 6: To the proof of Theorem 5.1, case III. a) fm(xm) = α(tm), fm(ym) = β(pm), 0 < tm, pm < 1 . III. b) The point z1 6∈ E. Since by the assumption ∂D ′ ⊂ E, the point z1 belongs to some component K1 of the set D ′ \ E. Since E is closed in Rn, it is closed with respect to D ′ . Therefore, K1 is an open set. Then the sequence fm(xm), which converges to z1 as m → ∞, itself belongs to K1 starting from some number,… view at source ↗
Figure 7
Figure 7. Figure 7: To the proof of Theorem 5.1, case III. b) be shown that the paths α and β may be chosen disjoint (if z1 = a1, this follows from the construction of these paths. If z1 6= a1, we first move from α its loops and apply Lemma 5.1. As a result we get a pair of disjoint paths, one of them is Jordanian, and another contains «start» and «end» of β, whenever β already contains the corresponding points fm(ym). Finall… view at source ↗
read the original abstract

The manuscript is devoted to the boundary behavior of mappings with bounded and finite distortion, which has been actively studied recently. We consider mappings of domains of the Euclidean space that satisfy the inverse Poletsky inequality with an integrable majorant, are open, and discrete. Assume that the image of the boundary of the original domain is finitely connected relative to the mapped domain, and the preimage of the boundary of the latter is nowhere a dense set. Then, under certain conditions on the geometry of these domains, it is proved that the specified mappings have a continuous boundary extension. The result is valid even in a more general form, when the majorant in the inverse Poletsky inequality is integrable over almost all concentric spheres centered at each point. In particular, the obtained results are valid for homeomorphisms as well as for open discrete closed mappings with the appropriate modulus condition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that open and discrete mappings of domains in Euclidean space satisfying the inverse Poletsky inequality with an integrable majorant admit continuous boundary extensions, under the hypotheses that the image of the boundary of the original domain is finitely connected relative to the target domain, the preimage of the target boundary is nowhere dense, and additional geometric conditions on the domains hold. The result is stated in a more general form where the majorant is integrable over almost all concentric spheres centered at each point, and it applies in particular to homeomorphisms as well as to open discrete closed mappings satisfying the modulus condition.

Significance. If the central theorem holds, the work extends Carathéodory-type boundary extension results to the setting of generalized quasiregular mappings (open discrete mappings with finite distortion) via modulus inequalities. The explicit geometric hypotheses and the sphere-integrability generalization provide a concrete, falsifiable framework that strengthens the applicability of modulus methods in geometric function theory.

minor comments (2)
  1. [Abstract] The abstract refers to 'certain conditions on the geometry of these domains' without naming them; a brief parenthetical list of the key geometric assumptions (e.g., finite connectivity, nowhere-dense preimage) would improve readability before the detailed statement in the introduction.
  2. [Introduction / Preliminaries] Notation for the inverse Poletsky inequality and the majorant function should be introduced with a displayed equation in the introduction or preliminaries section to make the central hypothesis immediately visible to readers familiar with modulus techniques.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a conditional theorem: under explicitly listed geometric hypotheses (finite connectivity of the image boundary relative to the target domain, nowhere-dense preimage of the target boundary) plus the inverse Poletsky inequality with integrable majorant, open discrete mappings admit continuous boundary extensions. These conditions are presented as assumptions rather than internally derived quantities. The derivation relies on standard modulus estimates and does not reduce any claimed prediction or uniqueness result to a self-definition, fitted input, or self-citation chain. The result is self-contained against external modulus theory and covers special cases (homeomorphisms, closed mappings) under the same stated modulus condition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard properties of modulus of curve families and openness/discreteness of mappings; no free parameters or invented entities are visible from the abstract.

axioms (2)
  • standard math Standard properties of the modulus of curve families in Euclidean space and the inverse Poletsky inequality
    Invoked throughout the statement of the main result
  • domain assumption Openness and discreteness of the mappings
    Explicitly assumed in the abstract

pith-pipeline@v0.9.0 · 5679 in / 1226 out tokens · 24421 ms · 2026-05-24T03:39:36.302663+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On Caratheodory theorem for open discrete unclosed mappings

    math.CV 2025-02 unverdicted novelty 4.0

    Equicontinuity of families of open discrete unclosed mappings satisfying inverse Poletsky inequalities is established via prime ends, yielding a result for Orlicz-Sobolev classes.

Reference graph

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