Vitushkin’s Conjecture for Removable Sets

Dudziak, James J.

doi:10.1007/978-1-4419-6709-1

Cited by 31 publications

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“…This result is an important step in the proof of Vitushkin's conjecture (for more details see [35,6]), which states that a compact set with finite one-dimensional Hausdorff measure is removable for bounded analytic functions if and only if it is purely 1-unrectifiable, which means that every 1-rectifiable subset of this set has Hausdorff measure zero. A higher dimensional analogue of Vitushkin's conjecture is proven in [24] but without using a higher dimensional version of Léger's theorem since in the higher dimensional setting there seems to be no connection between the n-dimensional Riesz transform and curvature (cf.…”

Section: Introductionmentioning

confidence: 92%

Integral Menger curvature and rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space

Meurer

2017

Trans. Amer. Math. Soc.

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Abstract. In this work we show that an n-dimensional Borel set in Euclidean N -space with finite integral Menger curvature is n-rectifiable, meaning that it can be covered by countably many images of Lipschitz continuous functions up to a null set in the sense of Hausdorff measure. This generalises Léger's [19] rectifiability result for one-dimensional sets to arbitrary dimension and co-dimension. In addition, we characterise possible integrands and discuss examples known from the literature.Intermediate results of independent interest include upper bounds of different versions of P. Jones's β-numbers in terms of integral Menger curvature without assuming lower Ahlfors regularity, in contrast to the results of Lerman and Whitehouse [20].

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Section: Introductionmentioning

confidence: 92%

Integral Menger curvature and rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space

Meurer

2017

Trans. Amer. Math. Soc.

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“…In [4] it was also noted that the estimate (3.1) is the best possible one for general compact sets, when the Hausdorff measure H 1 is replaced by the (possibly smaller) Hausdorff content H 1…”

Section: Painlevé Length Estimatesmentioning

confidence: 99%

Sharp estimate on the inner distance in planar domains

Lučić

Pasqualetto

Rajala

2020

Arkiv För Matematik

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We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlevé length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlevé length bound κ(E) ≤ πH 1 (E) is sharp.

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“…However, since K P has zero one dimensional Hausdorff measure one obtains that F extends conformally to the entire Riemann sphere by Painlevé's theorem (e.g. see [Dud10,Theorem 2.7]). Therefore f : P → Q is the restriction of a Möbius transformation.…”

Section: Standardnessmentioning

confidence: 99%

The Teichmüller space of the Hirsch foliation

Alvarez¹,

Lessa²

2018

Ann. inst. Fourier

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We prove that the Teichmüller space of the Hirsch foliation (a minimal foliation of a closed 3-manifold by non-compact hyperbolic surfaces) is homeomorphic to the space of closed curves in the plane. This allows us to show that that the space of hyperbolic metrics on the foliation is a trivial principal fiber bundle. And that the structure group of this bundle, the arc-connected component of the identity in the group of homeomorphisms which are smooth on each leaf and vary continuously in the smooth topology in the transverse direction of the foliation, is contractible.

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Vitushkin’s Conjecture for Removable Sets

Cited by 31 publications

References 0 publications

Integral Menger curvature and rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space

Integral Menger curvature and rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space

Sharp estimate on the inner distance in planar domains

The Teichmüller space of the Hirsch foliation

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