Jordan Curve Research Articles

Characterizing Meromorphic Pseudo-lemniscates

Let f be a meromorphic function with simply connected domain $$G\subset \mathbb {C}$$ , and let $$\Gamma \subset \mathbb {C}$$ be a smooth Jordan curve. We call a component of $$f^{-1}(\Gamma )$$ in G a $$\Gamma $$ -pseudo-lemniscate of f. In this note, we give criteria for a smooth Jordan curve $$\mathcal {S}$$ in G (with bounded face D) to be a $$\Gamma $$ -pseudo-lemniscate of f in terms of the number of preimages (counted with multiplicity) which a given w has under f in D (denoted $$\mathcal {N}_f(w)$$ ), as w ranges over the Riemann sphere. As a corollary, we obtain the fact that if $$\mathcal {N}_f(w)$$ takes three different value, then either $$\mathcal {S}$$ contains a critical point of f, or $$f(\mathcal {S})$$ is not a Jordan curve.

The Jordan Curve Theorem via Complex Analysis

AbstractThe aim of this article is to give a proof of the Jordan Curve Theorem via complex analysis.

Christoffel functions with power type weights

Precise asymptotics for Christoffel functions are established for power type weights on unions of Jordan curves and arcs. The asymptotics involve the equilibrium measure of the support of the measure. The result at the endpoints of arc components is obtained from the corresponding asymptotics for internal points with respect to a different power weight. On curve components the asymptotic formula is proved via a sharp form of Hilbert's lemniscate theorem while taking polynomial inverse images. The situation is completely different on the arc components, where the local asymptotics is obtained via a discretization of the equilibrium measure with respect to the zeros of an associated Bessel function. The proofs are potential theoretical, and fast decreasing polynomials play an essential role in them.

Harmonic measure on quasicircles and symmetric quasicircles

This paper is concerned with the study of harmonic measure on quasicircles and symmetric quasicircles. We investigate how the harmonic measure changes from one side of a Jordan curve to the other side. In particular, we characterize quasicircles and symmetric quasicircles using a type of harmonic symmetry property. We also explore the connection between quasicircles and the harmonic reflection property.

Removability and non-injectivity of conformal welding

We construct a (non-removable) Jordan curve $\Gamma$ and a non-M\{o}bius homeomorphism of the Riemann sphere which is conformal on the complement of $\Gamma$ and maps the curve $\Gamma$ onto itself. The curve is flexible in the sense of Bishop and may be taken to have zero area. The existence of such curves and conformal homeomorphisms is closely related to the non-injectivity of conformal welding.

Closure operators on graphs for modeling connectedness in digital spaces

For undirected simple graphs, we introduce closure operators on their vertex sets induced by sets of walks of the same lengths. Some basic properties of these closure operators are studied, with greater attention paid to connectedness. We focus on the closure operators induced by certain sets of walks in the 2-adjacency graph on the digital line Z, which generalize the Khalimsky topology. For the closure operators on Z2 obtained as particularly defined products of pairs of the induced closure operators on Z, we formulate and prove a digital form of the Jordan curve theorem.

Iteration of the Translated Tangent

The iteration of the map $$z \mapsto \lambda + \tan z$$ on the complex plane is investigated for each complex number $$\lambda $$ . It is shown that its Fatou set always contains a completely invariant component and the Julia set is either totally disconnected or connected. A necessary and sufficient condition for totally disconnected Julia set is proved. When the Julia set is connected, it is seen that the Fatou set can consist of additional components, which can be completely invariant attracting domain, a Siegel disk, a two periodic attracting domain or parabolic domain. Examples of each case are given. Further, a necessary and sufficient condition is provided for the Julia set to be a Jordan curve passing through infinity. It is proved that the set of parameters corresponding to Jordan curve Julia sets is connected, whereas the set of those corresponding to totally disconnected Julia sets is disconnected.

Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum

We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\infty $$\\end{document}. We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in C\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {C}}}{\\setminus }\\{0\\}$$\\end{document}. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the principal submatrices have to be real. The latter claim is also proved for matrices given by a more general symbol. The special role of the Jordan curve is further demonstrated by a new formula for the limiting density of the asymptotic eigenvalue distribution for banded Toeplitz matrices from the studied class. Certain connections between the problem under investigation, Jacobi operators, and the Hamburger moment problem are also discussed. The main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically.

Conformal trace theorem for Julia sets of quadratic polynomials

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

Nowhere differentiable functions of analytic type on products of finitely connected planar domains

Using Laurent decomposition and Mergelyan’s theorem combined with Baire’s category theorem, we prove generic nowhere differentiability on the distinguished boundary of functions of analytic type on products of planar domains bounded by finitely many disjoint Jordan curves. The parametrization of the boundaries are those induced by the natural Riemann maps.

Moduli space of cubic Newton maps

In this article, we study the topology and bifurcations of the moduli space M3 of cubic Newton maps. It's a subspace of the moduli space of cubic rational maps, carrying the Riemann orbifold structure (Cˆ,(2,3,∞)). We prove two results:• The boundary of the unique unbounded hyperbolic component is a Jordan arc and the boundaries of all other hyperbolic components are Jordan curves;• The Head's angle map is surjective and monotone. The fibers of this map are characterized completely.The first result is a moduli space analogue of the first author's dynamical regularity theorem [37]. The second result confirms a conjecture of Tan Lei.

Existence, regularity and structure of confined elasticae

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open setΩ. We prove existence, regularity and some structural properties of minimizers. In particular, whenΩis convex we show that a minimizer is necessarily a convex curve. We also provide an example of a minimizer with self-intersections.

Relation-induced connectedness in the digital plane

We introduce and discuss a connectedness induced by n-ary relations (\(n>1\) an integer) on their underlying sets. In particular, we focus on certain n-ary relations with the induced connectedness allowing for a definition of digital Jordan curves. For every integer \(n>1\), we introduce one such n-ary relation on the digital plane \({\mathbb {Z}}^2\) and prove a digital analogue of the Jordan curve theorem for the induced connectedness. It follows that these n-ary relations may be used as convenient structures on the digital plane for the study of geometric properties of digital images. For \(n=2\), such a structure coincides with the (specialization order of the) Khalimsky topology and, for \(n>2\), it allows for a variety of Jordan curves richer than that provided by the Khalimsky topology.

Abstract Voronoi Diagrams from Closed Bisecting Curves

We present the first algorithm for constructing abstract Voronoi diagrams from bisectors that are unbounded or closed Jordan curves. It runs in expected [Formula: see text] many steps and [Formula: see text] space, where [Formula: see text] is the number of sites, [Formula: see text] denotes the average number of faces (connected components) per Voronoi region in any diagram of a subset of [Formula: see text] sites, and [Formula: see text] is the maximum number of intersection points between any two related bisectors.

Sewing homeomorphism and conformal invariants

This paper is devoted to the study of some fundamental properties of the sewing homeomorphism induced by a Jordan domain. In particular, using conformal invariants such as harmonic measure, extremal distance, and reduced extremal distance, we give several necessary and sufficient conditions for the sewing homeomorphism to be bi-Lipschitz or bi-Holder. Furthermore, equivalent conditions for a Jordan curve to be a quasicircle are also obtained.

Embeddedness of the solutions to the H-Plateau problem

We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to constant mean curvature disks. We show that any minimizing H-disk in an H0-convex domain is embedded for any H∈[0,H0). In particular, for the unit ball B in R3, this implies that for any H∈[0,1], any Jordan curve in ∂B bounds an embedded H-disk in B.

Fast and Accurate Computation of the Logarithmic Capacity of Compact Sets

We present a numerical method for computing the logarithmic capacity of compact subsets of $\mathbb{C}$, which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have any special symmetry. The method relies on the conformal map onto lemniscatic domains and, computationally, on the solution of a boundary integral equation with the Neumann kernel. Our numerical examples indicate that the method is fast and accurate. We apply it to give an estimate of the logarithmic capacity of the Cantor middle third set and generalizations of it.

Nonlinear Riemann–Hilbert problems for quasilinear -equations on the unit disc

The existence of solutions of nonlinear Riemann–Hilbert problems for quasilinear -equations on the unit disc is considered. Let be a smooth function with bounded first derivatives and let be a family of Jordan curves in . Let be a smooth solution of the equation on such that belongs to the interior of the Jordan curve for every . Then there exists a smooth solution u of the equation on such that for every . Moreover, there is a sequence of solutions of this Riemann–Hilbert boundary value problem which uniformly on compact subsets of converges to .

New minimal surfaces in the hyperbolic space

We obtain new complete minimal surfaces in the hyperbolic space H3, by using Ribaucour transformations. Starting with the family of spherical catenoids in H3 found by Mori (1981), we obtain 2- and 3-parameter families of new minimal surfaces in the hyperbolic space, by solving a non trivial integro-differential system. Special choices of the parameters provide minimal surfaces whose parametrizations are defined on connected regions of R2 minus a disjoint union of Jordan curves. Any connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of H3 is a closed curve. The geometric properties of the surfaces regarding the ends, completeness and symmetries are discussed.

Radó–Kneser–Choquet theorem for harmonic mappings between surfaces

We simplify and improve a recent result of Martin (Trans AMS 368:647–658, 2016). Then we prove that if f is an orientation preserving harmonic mapping of the unit disk onto a $$C^{3,\alpha }$$ surface $$\Sigma $$ bounded by a Jordan curve $$\gamma \in C^{3,\alpha }$$ , that belongs to the boundary of a convex domain in $$\mathbf {R}^3$$ , then f is a diffeomorphism.