Hrant Hakobyan
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The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Hrant Hakobyan has a total of 11 co-authors. Here are the top co-authors by number of shared publications:
- Dragomir Šarić
Department of Mathematics, Queens College of CUNY, 65-30 Kissena Boulevard, Flushing, New York 11367; and Mathematics Ph.D. Program, The CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309
6 shared publications - Christopher J. Bishop
Stony Brook University, USA
2 shared publications - Marshall Williams
Department of Mathematics, Kansas State University, Manhattan, USA
2 shared publications Ara Basmajian
PhD Program in Mathematics The Graduate Center City University of New York New York New York USA
1 shared publication- Ara Basmajian
The Graduate Center, CUNY, 365 Fifth Ave., New York, New York 10016; and Hunter College, CUNY, 695 Park Ave., New York, New York 10065
1 shared publication Dragomir Šarić
PhD Program in Mathematics The Graduate Center City University of New York New York New York USA
1 shared publication
Latest Publications
- Open Access
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.Open AccessQuasisymmetric 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. - Open Access
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 $α_n$ 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 $\ell (α_n)$ of cuffs are at most $2\log n$, the surface $X$ is parabolic. One could expect that having arbitrary large cuff lengths $\ell (α_n)$ (think of $\ell (α_n)=n!^{n!}$) 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 Marković conjectured that for every choice of lengths $\ell (α_n)$, 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 $\ell (α_n)\geq a_n$ 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.Open AccessInducing 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 $α_n$ 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 $\ell (α_n)$ of cuffs are at most $2\log n$, the surface $X$ is parabolic. One could expect that having arbitrary large cuff lengths $\ell (α_n)$ (think of $\ell (α_n)=n!^{n!}$) 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 Marković conjectured that for every choice of lengths $\ell (α_n)$, 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 $\ell (α_n)\geq a_n$ 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. - Open Access
- 12 September 2023
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 376 (12) , 8877-8918
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 [Acta Math. 181 (1998), pp. 1–61]. 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 X can be associated to a “pillowcase sphere” X ^ \widehat {X} which is a metric space homeomorphic to the sphere S 2 \mathbb {S}^2 . We show that X X quasisymmetrically embeds into the plane if and only if X ^ \widehat {X} is quasisymmetric to S 2 \mathbb {S}^2 if and only if X ^ \widehat {X} is Ahlfors 2 2 -regular.
- 12 September 2023
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 376 (12) , 8877-8918
Open AccessQuasisymmetric 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 [Acta Math. 181 (1998), pp. 1–61]. 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 X can be associated to a “pillowcase sphere” X ^ \widehat {X} which is a metric space homeomorphic to the sphere S 2 \mathbb {S}^2 . We show that X X quasisymmetrically embeds into the plane if and only if X ^ \widehat {X} is quasisymmetric to S 2 \mathbb {S}^2 if and only if X ^ \widehat {X} is Ahlfors 2 2 -regular.
- Open Access
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 $\mathbb{T}^k$.Open AccessDivergent 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 $\mathbb{T}^k$. - Open Access
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.Open AccessThe 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. - 10 June 2022
- Published by Wiley in Proceedings of the London Mathematical Society
- Vol. 125 (3) , 568-625
The type problem for Riemann surfaces via Fenchel–Nielsen parameters
- 10 June 2022
- Published by Wiley in Proceedings of the London Mathematical Society
- Vol. 125 (3) , 568-625
The type problem for Riemann surfaces via Fenchel–Nielsen parameters
- 29 June 2021
- Published by Springer Nature in Journal d'Analyse Mathématique
- Vol. 143 (2) , 681-721
A Thurston boundary and visual sphere of the universal Teichmüller space
- 29 June 2021
- Published by Springer Nature in Journal d'Analyse Mathématique
- Vol. 143 (2) , 681-721
A Thurston boundary and visual sphere of the universal Teichmüller space
- Open Access
Quasisymmetric embeddings of slit Sierpiński carpets
We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpi\'nski 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.Open AccessQuasisymmetric embeddings of slit Sierpiński carpets
We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpi\'nski 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. - Open Access
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 Sierpi\'nski 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}.Open AccessQuasisymmetrically 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 Sierpi\'nski 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}. - 3 April 2018
- Published by Wiley in Proceedings of the London Mathematical Society
- Vol. 116 (6) , 1599-1628
Limits of Teichmüller geodesics in the Universal Teichmüller space
- 3 April 2018
- Published by Wiley in Proceedings of the London Mathematical Society
- Vol. 116 (6) , 1599-1628
Limits of Teichmüller geodesics in the Universal Teichmüller space