arXiv subjects

Dimitrios Ntalampekos

Publications and source records attributed to Dimitrios Ntalampekos.

32 records · Page 2Linked to original sources

arXiv2022

Conformal uniformization of planar packings by disk packings

A Sierpiński packing in the $2$-sphere is a countable collection of disjoint, non-separating continua with diameters shrinking to zero. We show that any Sierpiński packing by continua whose diameters are square-summable can be uniformized by a disk packing with a packing-conformal map, a notion that generalizes conformality in open sets. Being special cases of Sierpiński packings, Sierpiński carpets and some domains and can be uniformized by disk packings as well. As a corollary of the main result, the conformal loop ensemble (CLE) carpets can be uniformized conformally by disk packings, answering a question of Rohde-Werness.

math.CV↗

arXiv2022

Polyhedral approximation and uniformization for non-length surfaces

We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2-sphere with finite Hausdorff 2-measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala-Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk-Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain.

math.MG↗

arXiv2022

Polyhedral approximation of metric surfaces and applications to uniformization

We prove that any length metric space homeomorphic to a 2-manifold with boundary, also called a length surface, is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. As an application, using the classical uniformization theorem for Riemann surfaces and a limiting argument, we establish a general "one-sided" quasiconformal uniformization theorem for length surfaces with locally finite Hausdorff 2-measure. Our approach yields a new proof of the Bonk-Kleiner theorem characterizing Ahlfors 2-regular quasispheres.

math.MG↗

arXiv2021

On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

We construct quasiconformal mappings $f\colon \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ for which there is a Borel set $E \subset \mathbb{R}^2 \times \{0\}$ of positive Lebesgue $2$-measure whose image $f(E)$ has Hausdorff $2$-measure zero. This gives a solution to the open problem of inverse absolute continuity of quasiconformal mappings on hypersurfaces, attributed to Gehring. By implication, our result also answers questions of Väisälä and Astala--Bonk--Heinonen.

math.CV↗

arXiv2021

Extension of boundary homeomorphisms to mappings of finite distortion

We provide sufficient conditions so that a homeomorphism of the real line or of the circle admits an extension to a mapping of finite distortion in the upper half-plane or the disk, respectively. Moreover, we can ensure that the quasiconformal dilatation of the extension satisfies certain integrability conditions, such as $p$-integrability or exponential integrability. Mappings satisfying the latter integrability condition are also known as David homeomorphisms. Our extension operator is the same as the one used by Beurling and Ahlfors in their celebrated work. We prove an optimal bound for the quasiconformal dilatation of the Beurling--Ahlfors extension of a homeomorphism of the real line, in terms of its symmetric distortion function. More specifically, the quasiconformal dilatation is bounded above by an average of the symmetric distortion function and below by the symmetric distortion function itself. As a consequence, the quasiconformal dilatation of the Beurling--Ahlfors extension of a homeomorphism of the real line is (sub)exponentially integrable, is $p$-integrable, or has a $BMO$ majorant if and only if the symmetric distortion is (sub)exponentially integrable, is $p$-integrable, or has a $BMO$ majorant, respectively. These theorems are all new and reconcile several sufficient extension conditions that have been established in the past.

math.CV↗

arXiv2020

On the Hausdorff dimension of the residual set of a packing by smooth curves

Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has Hausdorff dimension bounded away from 1, quantitatively, depending only on the curvature bound. As a corollary, the residual set of any circle packing has Hausdorff dimension uniformly bounded away from 1. This result generalizes the result of Larman, who obtained the same conclusion for circle packings inside a square. We also show that our theorem is optimal and does not hold in general without lower curvature bounds. In particular, we construct packings by strictly convex, smooth curves whose residual sets have dimension 1. On the other hand, we prove that any packing by strictly convex curves cannot have $σ$-finite Hausdorff 1-measure.

math.CA↗

arXiv2019

Monotone Sobolev functions in planar domains: level sets and smooth approximation

We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded $1$-dimensional topological submanifold of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and minimum of the function in an open set are attained at the boundary. Our result is an analog of Sard's theorem, which asserts that for a $C^2$-smooth function in a planar domain almost every value is a regular value. As an application we show that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions.

math.CA↗

arXiv2019

Rigidity theorems for circle domains

A circle domain $Ω$ in the Riemann sphere is conformally rigid if every conformal map from $Ω$ onto another circle domain is the restriction of a Möbius transformation. We show that circle domains satisfying a certain quasihyperbolic condition, which was considered by Jones and Smirnov, are conformally rigid. In particular, Hölder circle domains and John circle domains are all conformally rigid. This provides new evidence for a conjecture of He and Schramm relating rigidity and conformal removability.

math.CV↗

arXiv2019

Non-removability of Sierpinski carpets

We prove that all Sierpiński carpets in the plane are non-removable for (quasi)conformal maps. More precisely, we show that for any two Sierpiński carpets $S,S'\subset \hat{\mathbb{C}}$ there exists a homeomorphism $f\colon \hat{\mathbb{C}}\to \hat{\mathbb{C}}$ that is conformal in $\hat{\mathbb{C}}\setminus S$ and it maps $S$ onto $S'$. The proof is topological and it utilizes the ideas of the topological characterization of Whyburn. As a corollary, we obtain a partial answer to a question of Bishop, whether any planar continuum with empty interior and positive measure can be mapped to a set of measure zero with an exceptional homeomorphism of the plane, conformal off that set.

math.CV↗

arXiv2019

A removability theorem for Sobolev functions and detour sets

We study the removability of compact sets for continuous Sobolev functions. In particular, we focus on sets with infinitely many complementary components, called "detour sets", which resemble the Sierpiński gasket. The main theorem is that if $K\subset \mathbb R^n$ is a detour set and its complementary components are sufficiently regular, then $K$ is $W^{1,p}$-removable for $p>n$. Several examples and constructions of sets where the theorem applies are given, including the Sierpiński gasket, Apollonian gaskets, and Julia sets.

math.CA↗

arXiv2019

Non-removability of Sierpinski spaces

We prove that all Sierpiński spaces in ${\mathbb{S}}^n$, $n\geq 2$, are non-removable for (quasi)conformal maps, generalizing the result of the first named author arXiv:1809.05605. More precisely, we show that for any Sierpiński space $X\subset \mathbb{S}^n$ there exists a homeomorphism $f\colon \mathbb{S}^n\to \mathbb{S}^n$, conformal in $\mathbb{S}^n\setminus X$, that maps $X$ to a set of positive measure and is not globally (quasi)conformal. This is the first class of examples of non-removable sets in higher dimensions.

math.MG↗

arXiv2018

Non-removability of the Sierpinski Gasket

We prove that the Sierpiński gasket is non-removable for quasiconformal maps, thus answering a question of Bishop. The proof involves a new technique of constructing an exceptional homeomorphism from $\mathbb R^2$ into some non-planar surface $S$, and then embedding this surface quasisymmetrically back into the plane by using the celebrated Bonk-Kleiner Theorem arXiv:math/0107171. 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 $1\leq p\leq 2$, thus complementing and sharpening the results of the author's previous work arXiv:1706.07687.

math.MG↗

arXiv2018

Potential theory on Sierpinski carpets with applications to uniformization

This research is motivated by the study of the geometry of fractal sets and is focused on uniformization problems: transformation of sets to canonical sets, using maps that preserve the geometry in some sense. More specifically, the main question addressed is the uniformization of planar Sierpinski carpets by square Sierpinski carpets, using methods of potential theory on carpets. We first develop a potential theory and study harmonic functions on planar Sierpinski carpets. We introduce a discrete notion of Sobolev spaces on Sierpinski carpets and use this to define harmonic functions. Our approach differs from the classical approach of potential theory in metric spaces because it takes the ambient space that contains the carpet into account. We prove basic properties such as the existence and uniqueness of the solution to the Dirichlet problem, Liouville's theorem, Harnack's inequality, strong maximum principle, and equicontinuity of harmonic functions. Then we utilize this notion of harmonic functions to prove a uniformization result for Sierpinski carpets. Namely, it is proved that every planar Sierpinski carpet whose peripheral disks are uniformly fat, uniform quasiballs can be mapped to a square Sierpinski carpet with a map that preserves carpet modulus. If the assumptions on the peripheral circles are strengthened to uniformly relatively separated, uniform quasicircles, then the map is a quasisymmetry. The real part of the uniformizing map is the solution of a certain Dirichlet-type problem. Then a harmonic conjugate of that map is constructed using the methods developed by Rajala arXiv:1412.3348.

math.MG↗

arXiv2017

Semi-hyperbolic rational maps and size of Fatou components

Recently Merenkov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpi${ń}$ski carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the Hausdorff dimension of the Julia set. We extend this theorem to Julia sets (not necessarily Sierpi${ń}$ski carpets) of semi-hyperbolic rational maps, and prove a stronger version of the theorem that was conjectured by Merenkov and Sabitova.

math.DS↗