Rigidity theorems for circle domains

Ntalampekos, Dimitrios; Younsi, Malik

doi:10.1007/s00222-019-00921-1

Cited by 15 publications

(15 citation statements)

References 33 publications

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“…The condition k Ω (•, x 0 ) ∈ L n (Ω) appeared in the work of Jones-Smirnov [JS00], where it is proved that this condition is sufficient for ∂Ω to be removable for quasiconformal homeomorphisms. This condition has also been used in recent work by the current author [Nta20] to establish the removability of certain fractals with infinitely many complementary components for Sobolev spaces; it also has appeared in work by the current author and Younsi [NY20] in establishing the rigidity of circle domains under this condition. We will use some auxiliary results from [NY20], which have been proved there in dimension 2, but the proofs apply to all dimensions.…”

Section: Examples Of Negligible Setsmentioning

confidence: 97%

“…A circle domain Ω is rigid if every conformal map from Ω onto another circle domain is the restriction of a Möbius transformation of C. He-Schramm [HS94] proved that circle domains whose boundary has σ-finite Hausdorff 1-measure are rigid. Later, Younsi and the author [NY20] proved the rigidity of circle domains with n-integrable quasihyperbolic distance (as in Theorem 1.5 (iii)). It is conjectured that rigidity of a circle domain is equivalent to removability of its boundary for quasiconformal homeomorphisms.…”

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confidence: 96%

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Definitions of quasiconformality and exceptional sets

Ntalampekos¹

2021

Preprint

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We present a new generalized metric definition of quasiconformality for Euclidean space, requiring that at each point there exists a sequence of uncentered open sets with bounded eccentricity shrinking to that point such that the images also have bounded eccentricity. This generalizes results of Gehring and Heinonen-Koskela on the metric definition of quasiconformality. We also study exceptional sets for this definition, in connection with sets that are negligible for extremal distance. We introduce the class of CNED sets, generalizing the notion of NED sets studied by Ahlfors-Beurling. A set E is CNED if the conformal modulus of a curve family is not affected when one restricts to the subfamily intersecting E at countably many points. We show as our main theorem that CNED sets are exceptional for the definition of quasiconformality. Our main theorem is an ultimate generalization of results of Gehring, Heinonen-Koskela, and Kallunki-Koskela. We prove that several classes of sets that were known to be exceptional are also CNED, including sets of σ-finite Hausdorff (n − 1)-measure and boundaries of domains with n-integrable quasihyperbolic distance. Thus, this work puts in common framework many known results on the problem of quasiconformal removability and provides evidence that the CNED condition should also be necessary for removability. We prove that countable unions of closed (C )NED sets are (C )NED, and therefore we enlarge significantly the known classes of quasiconformally removable sets. On the other hand, we present examples showing that unions of non-closed negligible or exceptional sets need not be negligible or exceptional, respectively. Finally, we give an application to the problem of rigidity of circle domains.

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Section: Examples Of Negligible Setsmentioning

confidence: 97%

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confidence: 96%

Definitions of quasiconformality and exceptional sets

Ntalampekos¹

2021

Preprint

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“…It suffices to show that if j is large enough, then (18) mod(S(z, r) \ p j , S(z, R); D j ) ǫ(r) → 0 as r → 0, where ǫ(r) does not depend on j. We will do this by first constructing a suitable sequence of disjoint annuli, and then applying them to find admissible functions.…”

Section: 3mentioning

confidence: 99%

“…See also [11], [12], [13]. Recently, results related to the Koebe conjecture have been established in [2], [14], [16], [18], [21], and [22].…”

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Uniformization of planar domains by exhaustion

Rajala¹

2021

Preprint

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We study the method of finding conformal maps onto circle domains by approximating with finitely connected subdomains. Every domain D ⊂ Ĉ admits exhaustions, i.e., increasing sequences of finitely connected subdomains Dj whose union is D. By Koebe's theorem, each Dj admits a conformal map fD j from Dj onto a circle domain fD j (Dj ). Assuming fD j → f , our goal is to find out if f (D) is also a circle domain.We present a countably connected D with an exhaustion (Dj ) so that (fD j ) has a limit whose image is not a circle domain, and a domain Ω with an exhaustion (Ωj ) so that (fΩ j ) has a limit whose image has uncountably many non-point complementary components.On the other hand, we prove that every exhaustion (Dj) of a countably connected D admits a refinement so that the image of the corresponding limit map is a circle domain. Our result extends the He-Schramm theorem on the uniformization of countably connected domains and provides a new proof.

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“…Moreover, further connections between the problems of rigidity of circle domains and removability of their boundary have been established in [NY18], in the spirit of the conjecture of He and Schramm [HS94]. In particular, it is proved that all circle domains satisfying the quasihyperbolic condition of [JS00] are rigid.…”

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Non-removability of the Sierpiński gasket

Ntalampekos

2019

Invent. math.

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We prove that the Sierpiński gasket is non-removable for quasiconformal maps, thus answering a question of Bishop [Bi15]. The proof involves a new technique of constructing an exceptional homeomorphism from R 2 into some non-planar surface S, and then embedding this surface quasisymmetrically back into the plane by using the celebrated Bonk-Kleiner Theorem [BK02]. We also prove that all homeomorphic copies of the Sierpiński gasket are non-removable for continuous Sobolev functions of the class W 1,p for 1 ≤ p ≤ 2, thus complementing and sharpening the results of the author's previous work [Nt17].

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Rigidity theorems for circle domains

Cited by 15 publications

References 33 publications

Definitions of quasiconformality and exceptional sets

Definitions of quasiconformality and exceptional sets

Uniformization of planar domains by exhaustion

Non-removability of the Sierpiński gasket

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