arXiv subjects

Hrant Hakobyan

Publications and source records attributed to Hrant Hakobyan.

12 recordsLinked to original sources

arXiv2025

Quasisymmetric Koebe Uniformization of metric surfaces

We study when a metric surface $X$ can be mapped quasisymmetrically onto a circle domain $D\subset\mathbb{C}$ with uniformly relatively separated boundary components. Bonk \cite{Bonk} proved that if $X\subset \hat{\mathbb{C}}$ and the boundary components of $X$ are uniformly relatively separated uniform quasicircles then $X$ is quasisymmetric to a circle domain. Merenkov and Wildrick \cite{Merenkov Wildrick} showed that Bonk's condition is not sufficient in the non-planar case. We prove that under some mild assumptions, a metric surface is quasisymmetric to a circle domain with uniformly relatively separated boundary components if and only if it is 2-TLP. The latter is a version of a condition introduced and studied by Bonk \cite{Bonk}. This answers a question of Merenkov and Wildrick in \cite{Merenkov Wildrick} and it is also a natural generalization of Bonk's result to non-planar metric surfaces.

math.CV↗

arXiv2024

Inducing recurrent flows by twisting on infinite surfaces with unbounded cuffs

A Riemann surface $X$ is parabolic if and only if the geodesic flow (for the hyperbolic metric) on the unit tangent bundle of $X$ is ergodic. Consider a Riemann surface $X$ with a single topological end and a sequence $\alpha_n$ of pairwise disjoint, simple closed geodesics converging to the end, called {\it cuffs}. Basmajian, the first and the third author, proved that when the lengths $\ell (\alpha_n)$ of cuffs are at most $2\log n$, the surface $X$ is parabolic. One could expect that having arbitrary large cuff lengths $\ell (\alpha_n)$ (think of $\ell (\alpha_n)=n!^{n!}$) would allow the geodesic flow to escape to infinity, thus making $X$ not parabolic. Contrary to this and motivated by their proof of the Surface Subgroup Theorem, Kahn and Markovi\'c conjectured that for every choice of lengths $\ell (\alpha_n)$, there is a choice of twists that would make $X$ parabolic. We show that their conjecture is essentially true. Namely, for any sequence of positive numbers $\{ a_n\}$, there is a choice of lengths $\ell (\alpha_n)\geq a_n$ such that the (relative) twists by $1/2$ make $X$ parabolic. This result extends to the surfaces with countably many ends while it does not hold for uncountably many ends.

math.DS↗

arXiv2023

Conformal Dimension of the Brownian Graph

Conformal dimension of a metric space $X$, denoted by $\dim_C X$, is the infimum of the Hausdorff dimension among all its quasisymmetric images. If conformal dimension of $X$ is equal to its Hausdorff dimension, $X$ is said to be minimal for conformal dimension. In this paper we show that the graph of the one dimensional Brownian motion is almost surely minimal for conformal dimension. We also give many other examples of minimal sets for conformal dimension, which we call Bedford-McMullen type sets. In particular we show that Bedford-McMullen self-affine sets with uniform fibers are minimal for conformal dimension. The main technique in the proofs is the construction of ``rich families of minimal sets of conformal dimension one''. The latter concept is quantified using Fuglede's modulus of measures.

math.MG↗

arXiv2023

Divergent geodesics in the Universal Teichmüller space

Thurston boundary of the universal Teichmüller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of generalized Teichmüller type if it shrinks the vertical foliation of a holomorphic quadratic differential. We provide the first examples of generalized Teichmüller rays which diverge near Thurston boundary $PLM_{bdd}(\mathbb{D})$. Moreover, for every $k\geq 1$ we construct examples of rays with limit sets homeomorphic to $k$-dimensional cubes. For the latter result we utilize the classical Kronecker approximation theorem from number theory which states that if $θ_1,\ldots,θ_k$ are rationally independent reals then the sequence $(\{θ_1 n\},\ldots,\{θ_k n\})$ is dense in the $k$-torus $\mathbb{T}^k$.

math.CV↗

arXiv2022

The type problem for Riemann surfaces via Fenchel-Nielsen parameters

A Riemann surface $X$ is said to be of \emph{parabolic type} if it supports a Green's function. Equivalently, the geodesic flow on the unit tangent of $X$ is ergodic. Given a Riemann surface $X$ of arbitrary topological type and a hyperbolic pants decomposition of $X$ we obtain sufficient conditions for parabolicity of $X$ in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of \textit{non standard half-collar} about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two non standard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, Abelian covers of compact surfaces are also studied.

math.GT↗

arXiv2021

Quasisymmetric embeddings of slit Sierpiński carpets

We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpiński carpet into the plane. In the case of so called dyadic slit carpets, several characterizations are obtained. One characterization is in terms of a Transboundary Loewner Property (TLP) which is a transboundary analogue of the Loewner property of Heinonen and Koskela. We show that a dyadic slit carpet can be quasisymmetrically embedded into the plane if and only if it is TLP. Moreover, every dyadic slit carpet $X$ can be associated to a "pillowcase sphere" $\widehat{X}$ which is a metric space homeomorphic to the sphere $\mathbb{S}^2$. We show that $X$ quasisymmetrically embeds into the plane if and only if $\widehat{X}$ is quasisymmetric to $\mathbb{S}^2$ if and only if $\widehat{X}$ is Ahlfors $2$-regular.

math.MG↗

arXiv2018

Quasisymmetrically co-Hopfian Sierpiński Spaces and Menger Curve

A metric space $X$ is quasisymmetrically co-Hopfian if every quasisymmetric embedding of $X$ into itself is onto. We construct the first examples of metric spaces homeomorphic to the universal Menger curve and higher dimensional Sierpiński spaces, which are quasisymmetrically co-Hopfian. We also show that the collection of quasisymmetric equivalence classes of spaces homeomorphic to the Menger curve is uncountable. These results answer a problem and generalize results of Merenkov from \cite{Mer:coHopf}.

math.CV↗

arXiv2016

Qusisymmetric dimension distortion of Ahlfors regular subsets of a metric space

We show that if $f:X\to Y$ is a quasisymmetric mapping between Ahlfors regular spaces, then $\dim_H f(E)\leq\dim_H E$ for "almost every" bounded Ahlfors regular set $E\subseteq X$. If additionally, $X$ and $Y$ are Loewner spaces then $\dim_H f(E)=\dim_H E$ for "almost every" Ahlfors regular set $E\subset X$. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if $f$ is a quasiconformal map of $\mathbb{R}^N$, $N\geq 2$, then for Lebesgue a.e. $y\in\mathbb{R}^N$ we have $\dim_H f(y+E) = \dim_H E$. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if $E \subset \mathbb{R}$ is Ahlfors $d$-regular, $d<1$, then some component of $f(E \times \mathbb{R})$ has dimension at most $2/(d+1)$, and we construct examples to show this bound is sharp. In addition, we show there is a $1$-dimensional set $S\subseteq \mathbb R$ and planar quasiconformal map $f$ such that $f(\mathbb{R} \times S)$ contains no rectifiable sub-arcs. These results generalize work of Balogh, Monti and Tyson \cite{Tyson:frequency} and answer questions posed in \cite{Tyson:frequency} and \cite{AimPL}.

math.CV↗

arXiv2015

Visual sphere and Thurston's boundary of the Universal Teichmüller space

Thurston's boundary to the universal Teichmüller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of Teichmüller type if it shrinks vertical foliation of an integrable holomorphic quadratic differential. In a prior work we established that each Teichmüller geodesic ray limits to a multiple (by the reciprocal of the length of the leaves) of vertical foliation of the quadratic differential. Certain non-integrable holomorphic quadratic differential induce geodesic rays and we consider their limit points in $PML_{bdd}(\mathbb{D})$. Somewhat surprisingly, the support of the limiting projective measured laminations might be a geodesic lamination whose leaves are not homotopic to leaves of either vertical or horizontal foliation of the non-integrable holomorphic quadratic differential.

math.GT↗

arXiv2015

Limits of Teichmüller geodesics in the Universal Teichmüller space

Thurston's boundary to the universal Teichmüller space $T(\mathbb{H})$ is the set of asymptotic rays to the embedding of $T(\mathbb{H})$ in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations $PML_{bdd}(\mathbb{H})$ of $\mathbb{H}$. We prove that each Teichmüller geodesic ray in $T(\mathbb{H})$ has a unique limit point in Thurston's boundary to $T(\mathbb{H})$ unlike in the case of closed surfaces.

math.CV↗

arXiv2014

Vertical Limits of Graph Domains

We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space $T(\mathbb{H})$. Our main result states that the limits of the Teichmüller geodesics in the Thurston's boundary of $T(\mathbb{H})$ may depend on both vertical and horizontal foliation of the corresponding holomorphic quadratic differential.

math.CV↗

arXiv2008

Conformal dimension: Cantor sets and moduli

In this paper we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1 thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede \cite{Fug}. It implies in particular that there are many sets $E\subset\mathbb{R}$ of zero length such that $X\times Y$ is minimal for conformal dimension for every compact $Y$.

math.CV↗