Dragomir Šarić
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Dragomir Šarić has a total of 19 co-authors. Here are the top co-authors by number of shared publications:
- Hrant Hakobyan
Department of Mathematics, Kansas State University, Manhattan, USA
6 shared publications - 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
3 shared publications - Francis Bonahon
Department of Mathematics, University of Southern California, Los Angeles, USA
2 shared publications - Hideki Miyachi
Osaka University
2 shared publications - Hideki Miyachi
School of Mathematics and Physics, College of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, Japan
2 shared publications - Michael Pandazis
PhD Program in Mathematics, The Graduate Center, CUNY, 365 Fifth Ave., New York, New York 10016
2 shared publications
Latest Publications
- Open Access
- 5 January 2026
- Published by Finnish Mathematical Society in Annales Fennici Mathematici
- Vol. 51 (1) , 3-16
Teichmüller extremal maps on infinite Riemann surfaces
Let \(X=\mathbb{D}/\Gamma\) be a Riemann surface with \(\Gamma\) of the first kind. We establish a necessary and sufficient criterion for \([f]\in T(X)\) to have a Teichmüller-type extremal map.- 5 January 2026
- Published by Finnish Mathematical Society in Annales Fennici Mathematici
- Vol. 51 (1) , 3-16
Open AccessTeichmüller extremal maps on infinite Riemann surfaces
Let \(X=\mathbb{D}/\Gamma\) be a Riemann surface with \(\Gamma\) of the first kind. We establish a necessary and sufficient criterion for \([f]\in T(X)\) to have a Teichmüller-type extremal map. - Open Access
Intersection numbers between horizontal foliations of quadratic differentials
We establish that the intersection number between the horizontal foliations of any two finite-area holomorphic quadratic differentials on an arbitrary Riemann surface is finite. Our main result shows that the intersection number is jointly continuous in the $L^1$-norm on the quadratic differentials. A corollary is that the Jenkins-Strebel differentials are not dense in the space of all finite-area holomorphic quadratic differentials when the infinite Riemann surface is not parabolic.
Open AccessIntersection numbers between horizontal foliations of quadratic differentials
We establish that the intersection number between the horizontal foliations of any two finite-area holomorphic quadratic differentials on an arbitrary Riemann surface is finite. Our main result shows that the intersection number is jointly continuous in the $L^1$-norm on the quadratic differentials. A corollary is that the Jenkins-Strebel differentials are not dense in the space of all finite-area holomorphic quadratic differentials when the infinite Riemann surface is not parabolic.
- Open Access
Bounded ideal triangulations of infinite Riemann surfaces
We introduce a notion of a bounded ideal triangulation of an infinite Riemann surface and parametrize Teichmüller spaces of infinite surfaces which allow bounded triangulations. We prove that our parametrization is real-analytic. Riemann surfaces with bounded geometry and countably many punctures belong to the class of surfaces with bounded ideal triangulations. In comparison, the Fenchel-Nielsen parametrization for surfaces with bounded geometry is not known, while the Fenchel-Nielsen parametrization for surfaces with bounded pants decompositions is known as a homeomorphism but it is not known whether it is real-analytic
Open AccessBounded ideal triangulations of infinite Riemann surfaces
We introduce a notion of a bounded ideal triangulation of an infinite Riemann surface and parametrize Teichmüller spaces of infinite surfaces which allow bounded triangulations. We prove that our parametrization is real-analytic. Riemann surfaces with bounded geometry and countably many punctures belong to the class of surfaces with bounded ideal triangulations. In comparison, the Fenchel-Nielsen parametrization for surfaces with bounded geometry is not known, while the Fenchel-Nielsen parametrization for surfaces with bounded pants decompositions is known as a homeomorphism but it is not known whether it is real-analytic
- 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
On the derivatives of the Liouville currents
The Liouville map, introduced by Bonahon, assigns to each point in the Teichmüller space a natural Radon measure on the space of geodesics of the base surface. The Liouville map is real analytic and it even extends to a holomorphic map of a neighborhood of the Teichmüller space in the Quasi-Fuchsian space of an arbitrary conformally hyperbolic Riemann surface. The earthquake paths and by their extension quake-bends, introduced by Thurston, are particularly nice real-analytic and holomorphic paths in the Teichmüller and the Quasi-Fuchsian space, respectively. We find a geometric expression for the derivative of the Liouville map along earthquake paths.
Open AccessOn the derivatives of the Liouville currents
The Liouville map, introduced by Bonahon, assigns to each point in the Teichmüller space a natural Radon measure on the space of geodesics of the base surface. The Liouville map is real analytic and it even extends to a holomorphic map of a neighborhood of the Teichmüller space in the Quasi-Fuchsian space of an arbitrary conformally hyperbolic Riemann surface. The earthquake paths and by their extension quake-bends, introduced by Thurston, are particularly nice real-analytic and holomorphic paths in the Teichmüller and the Quasi-Fuchsian space, respectively. We find a geometric expression for the derivative of the Liouville map along earthquake paths.
- 9 February 2024
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 306 (3) , 1-14
Quasisymmetric maps, shears, lambda lengths and flips
- 9 February 2024
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 306 (3) , 1-14
Quasisymmetric maps, shears, lambda lengths and flips
- Open Access
Circle homeomorphisms with square summable diamond shears
We introduce and the study the space of homeomorphisms of the circle (up to M\"obius transformations) which are in $\ell^2$ with respect to modular coordinates called diamond shears along the edges of the Farey tessellation. Diamond shears are related combinatorially to shear coordinates, and are also closely related to the $\log \Lambda$-lengths of decorated Teichm\"uller space introduced by Penner. We obtain sharp results comparing this new class to the Weil-Petersson class and H\"older classes of circle homeomorphisms. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.Open AccessCircle homeomorphisms with square summable diamond shears
We introduce and the study the space of homeomorphisms of the circle (up to M\"obius transformations) which are in $\ell^2$ with respect to modular coordinates called diamond shears along the edges of the Farey tessellation. Diamond shears are related combinatorially to shear coordinates, and are also closely related to the $\log \Lambda$-lengths of decorated Teichm\"uller space introduced by Penner. We obtain sharp results comparing this new class to the Weil-Petersson class and H\"older classes of circle homeomorphisms. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears. - Open Access
Quadratic differentials and foliations on infinite Riemann surfaces
We prove that an infinite Riemann surface $X$ is parabolic ($X\in O_G$) if and only if the union of the horizontal trajectories of any integrable holomorphic quadratic differential that are cross-cuts is of zero measure. Then we establish the density of the Jenkins-Strebel differentials in the space of all integrable quadratic differentials when $X\in O_G$ and extend Kerckhoff's formula for the Teichm\"uller metric in this case. Our methods depend on extending to infinite surfaces the Hubbard-Masur theorem describing which measured foliations can be realized by horizontal trajectories of integrable holomorphic quadratic differentials.Open AccessQuadratic differentials and foliations on infinite Riemann surfaces
We prove that an infinite Riemann surface $X$ is parabolic ($X\in O_G$) if and only if the union of the horizontal trajectories of any integrable holomorphic quadratic differential that are cross-cuts is of zero measure. Then we establish the density of the Jenkins-Strebel differentials in the space of all integrable quadratic differentials when $X\in O_G$ and extend Kerckhoff's formula for the Teichm\"uller metric in this case. Our methods depend on extending to infinite surfaces the Hubbard-Masur theorem describing which measured foliations can be realized by horizontal trajectories of integrable holomorphic quadratic differentials. - Open Access
On complex extension of the Liouville map
The Liouville map assigns to each point in the Teichm\"uller space a positive Radon measure on the space of geodesics of the universal covering of the base Riemann surface. This construction which was introduced by Bonahon is valid for both finite and infinite Riemann surfaces. Bonahon and S\"ozen proved that the Liouville map is differentiable for closed Riemann surfaces and the second author extended this result to all other Riemann surfaces. Otal proved that the Liouville map is real analytic using an idea from the geometric analysis. The purpose of this note is to give another proof of Otal's result using a complex analysis approach.
Open AccessOn complex extension of the Liouville map
The Liouville map assigns to each point in the Teichm\"uller space a positive Radon measure on the space of geodesics of the universal covering of the base Riemann surface. This construction which was introduced by Bonahon is valid for both finite and infinite Riemann surfaces. Bonahon and S\"ozen proved that the Liouville map is differentiable for closed Riemann surfaces and the second author extended this result to all other Riemann surfaces. Otal proved that the Liouville map is real analytic using an idea from the geometric analysis. The purpose of this note is to give another proof of Otal's result using a complex analysis approach.
- 19 May 2023
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 376 (10) , 7013-7043
Ergodicity of the geodesic flow on symmetric surfaces
We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X X that guarantee the surface X X is of parabolic type. An interesting class of Riemann surfaces for this problem is the one with finitely many topological ends. In this case the length part of the Fenchel-Nielsen coordinates can go to infinity for parabolic X X . When the surface X X is end symmetric, we prove that X X being parabolic is equivalent to the covering group being of the first kind. Then we give necessary and sufficient conditions on the Fenchel-Nielsen coordinates of a half-twist symmetric surface X X such that X X is parabolic. As an application, we solve an open question from the prior work of Basmajian, Hakobyan and the second author [Proc. Lond. Math. Soc. (3) 125 (2022), pp. 568–625].
- 19 May 2023
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 376 (10) , 7013-7043
Ergodicity of the geodesic flow on symmetric surfaces
We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X X that guarantee the surface X X is of parabolic type. An interesting class of Riemann surfaces for this problem is the one with finitely many topological ends. In this case the length part of the Fenchel-Nielsen coordinates can go to infinity for parabolic X X . When the surface X X is end symmetric, we prove that X X being parabolic is equivalent to the covering group being of the first kind. Then we give necessary and sufficient conditions on the Fenchel-Nielsen coordinates of a half-twist symmetric surface X X such that X X is parabolic. As an application, we solve an open question from the prior work of Basmajian, Hakobyan and the second author [Proc. Lond. Math. Soc. (3) 125 (2022), pp. 568–625].