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Malik Younsi

Publications and source records attributed to Malik Younsi.

At least 19 recordsLinked to original sources
arXiv2026

Holomorphic motions, Assouad dimension and quasiconformal mappings

We study the variation of the quasi-Assouad dimension of a set moving under a holomorphic motion. We show that the reciprocal of the quasi-Assouad dimension is inf-harmonic in the sense of Fuhrer--Ransford--Younsi. As a consequence, we obtain quasiconformal distortion bounds for quasi-Assouad dimension as well as an improved version of Smirnov's celebrated theorem on the dimension of quasicircles. Our approach is elementary in that it does not require optimal Sobolev regularity for quasiconformal mappings.

math.CV↗
arXiv2025

Rational lemniscates and the matching problem

The matching problem for a given Jordan curve in the complex plane asks to find two nonconstant functions, one analytic in the bounded complementary component of the curve and the other analytic in the unbounded complementary component of the curve, which are continuous up to the curve and complex conjugate to each other on the curve. We prove that there exist Jordan curves of any Hausdorff dimension between $1$ and $2$ for which the matching problem has a solution. This answers a question of Ebenfelt--Khavinson--Shapiro and provides the first examples of solutions to the matching problem other than rational lemniscates. Our approach relies on conformal welding and harmonic measure. We also obtain new examples of Jordan curves for which the matching problem has no solution, and give a characterization of the subsets of the Riemann sphere that are rational lemniscates in terms of harmonic measure.

math.CV↗
arXiv2025

Continuous analytic capacity and holomorphic motions

We construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion, thereby answering a question of Paul Gauthier. Our example is inspired by holomorphic dynamics and relies on the works of Bishop--Carleson--Garnett--Jones and Browder--Wermer relating tangent points of Jordan curves, harmonic measure and Dirichlet algebras. Our approach also provides a new proof of a result of Ransford, Younsi and Ai on the variation of analytic capacity under holomorphic motions. In addition, we show that extremal functions for continuous analytic capacity may not exist.

math.CV↗
arXiv2023

Holomorphic motions, dimension, area and quasiconformal mappings

We describe the variation of the Minkowski, packing and Hausdorff dimensions of a set moving under a holomorphic motion, as well as the variation of its area. Our method provides a new, unified approach to various celebrated theorems about quasiconformal mappings, including the work of Astala on the distortion of area and dimension under quasiconformal mappings and the work of Smirnov on the dimension of quasicircles.

math.CV↗
arXiv2020

Analytic capacity and holomorphic motions

We study the behavior of the analytic capacity of a compact set under deformations obtained by families of conformal maps depending holomorphically on the complex parameter. We show that, under those deformations, the logarithm of the analytic capacity varies harmonically. We also show that the hypotheses in this result cannot be substantially weakened.

math.CV↗
arXiv2020

Continuity of capacity of a holomorphic motion

We study the behavior of various set-functions under holomorphic motions. We show that, under such deformations, logarithmic capacity varies continuously, while analytic capacity may not.

math.CV↗
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↗
arXiv2018

Computing polynomial conformal models for low-degree Blaschke products

For any finite Blaschke product $B$, there is an injective analytic map $φ:\mathbb{D}\to\mathbb{C}$ and a polynomial $p$ of the same degree as $B$ such that $B=p\circφ$ on $\mathbb{D}$. Several proofs of this result have been given over the past several years, using fundamentally different methods. However, even for low-degree Blaschke products, no method has hitherto been developed to explicitly compute the polynomial $p$ or the associated conformal map $φ$. In this paper, we show how these functions may be computed for a Blaschke product of degree at most three, as well as for Blaschke products of arbitrary degree whose zeros are equally spaced on a circle centered at the origin.

math.CV↗
arXiv2017

Fekete polynomials and shapes of Julia sets

We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation in terms of geometric and potential theoretic quantities.

math.CV↗
arXiv2017

Removability and non-injectivity of conformal welding

We construct a (non-removable) Jordan curve $Γ$ and a non-Möbius homeomorphism of the Riemann sphere which is conformal on the complement of $Γ$ and maps the curve $Γ$ 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.

math.CV↗
arXiv2016

Conformal models and fingerprints of pseudo-lemniscates

We prove that every function that is meromorphic on the closure of an analytic Jordan domain and sufficiently well-behaved on the boundary is conformally equivalent to a rational map whose degree is smallest possible. We also show that the minimality of the degree fails in general without the boundary assumptions. As an application, we generalize a theorem of Ebenfelt, Khavinson and Shapiro by characterizing fingerprints of polynomial pseudo-lemniscates.

math.CV↗
arXiv2015

Removability, rigidity of circle domains and Koebe's Conjecture

A circle domain $Ω$ in the Riemann sphere is conformally rigid if every conformal map of $Ω$ onto another circle domain is the restriction of a Möbius transformation. We show that two rigidity conjectures of He and Schramm are in fact equivalent, at least for a large family of circle domains. The proof follows from a result on the removability of countable unions of certain conformally removable sets. We also introduce trans-quasiconformal deformation of Schottky groups to prove that a circle domain is conformally rigid if and only if it is quasiconformally rigid, thereby providing new evidence for the aforementioned conjectures.

math.CV↗
arXiv2015

On removable sets for holomorphic functions

We present a comprehensive survey on removability of compact plane sets with respect to various classes of holomorphic functions. We also discuss some applications and several open questions, some of which are new.

math.CV↗
arXiv2014

On the analytic and Cauchy capacities

We give new sufficient conditions for a compact set $E \subseteq \mathbb{C}$ to satisfy $γ(E)=γ_c(E)$, where $γ$ is the analytic capacity and $γ_c$ is the Cauchy capacity. As a consequence, we provide examples of compact plane sets such that the above equality holds but the Ahlfors function is not the Cauchy transform of any complex Borel measure supported on the set.

math.CV↗
arXiv2014

Shapes, fingerprints and rational lemniscates

It has been known since the work of A.A. Kirillov that any smooth Jordan curve in the plane can be represented by its so-called fingerprint, an orientation preserving smooth diffeomorphism of the unit circle onto itself. In this paper, we give a new, simple proof of a theorem of Ebenfelt, Khavinson and Shapiro stating that the fingerprint of a polynomial lemniscate of degree $n$ is given by the $n$-th root of a Blaschke product of degree $n$ and that conversely, any smooth diffeomorphism induced by such a map is the fingerprint of a polynomial lemniscate of the same degree. The proof is easily generalized to the case of rational lemniscates, thus solving a problem raised by the previously mentioned authors.

math.CV↗
arXiv2014

Rational Ahlfors Functions

We study a problem of Jeong and Taniguchi asking to find all rational maps which are Ahlfors functions. We prove that the rational Ahlfors functions of degree two are characterized by having positive residues at their poles. We then show that this characterization does not generalize to higher degrees, with the help of a numerical method for the computation of analytic capacity. We also provide examples of rational Ahlfors functions in all degrees.

math.CV↗
arXiv2013

Finitely connected domains, Rational maps and Ahlfors functions

Using Ahlfors functions, Grunsky maps and the Bell representation theorem, we show that a certain subset of the rational maps of degree $n$ forms a trivial bundle over the moduli space of non-degenerate $n$-connected domains with one marked tangent vector with fiber the $n$-fold symmetric product of the circle. A consequence is that the set of rational Ahlfors functions of degree $n$ forms a closed embedded submanifold inside the space of rational maps of degree $n$. As an application, we show the existence of rational Ahlfors functions with non-positive residues, resolving a question left open in a previous paper by the authors.

math.CV↗