arXiv research

Defines horocyclic evolutes, parallels, and involutes of spacelike frontals in hyperbolic 2-space.

New surfaces with special geodesic and horocycle behaviors discovered.

Non-ergodic measures found in horocycle flow on Abelian differentials.

A bijection preserving horocycles/hypercycles is an isometry in hyperbolic plane.

Study horocycle orbits in $\mathbb{Z}$-covers of hyperbolic surfaces.

Study geometric actions of groups on horocyclic products.

Classifies horocycle flow closures in hyperbolic 3-manifolds.

The paper proves inequalities for hyperbolic sets and curves.

Pack hyperbolic surfaces with circles or horocycles, noting symmetries.

A support theorem for the horocycle Radon transform f \to \hat{f} is a property of the form \hat{f} of compact support \rightarrow f of compact support. Here we prove a variation of this result where support (\hat{f}) is outside a fixed horocycle in hyperbolic space.

New surfaces show horocyclic flow isn't always minimal.

We introduce a deformation of Riemann surfaces and we are interested in the convergence of this deformation to a point of the Gardiner-masur boundary of Teichmueller space. This deformation, which we call the horocyclic deformation, is directed by a projective measured foliation and belongs to a certain horocycle in a …

Using zippered rectangle coordinates we parametrize a Poincaré section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle $6π$. In addition, we bound the return time under horocycle flow to this Poincaré section by examining a subset of surfaces where a certain sum of …

We give an example of a horocycle in the Teichmüller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichmüller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is…

Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $Ω$ be a 2-form. We show that the magnetic flow of the pair $(g,Ω)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $Ω$ is a constant multiple of the area form of $g$ and the mag…

In this work we investigate the following isoperimetric problem in the hyperbolic plane: to find the regions of prescribed area with minimal perimeter between two parallel horocycles. We give an explicit and detailed description of all such regions.

We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…

We consider flows, called $W^{\rm u}$ flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of $W^{\rm u}$ flows and we show that $W^{\rm u}$ flows have purely absolutely continuous spectrum in the orthocom…

The paper defines catenary curves in spheres and hyperbolic planes.

Study CMC hypersurfaces in $\mathbb{H}^2 imes\mathbb{H}^2$ with a specific symmetry.

Proof of Graustein's theorem in different geometries.

The horocyclic flow on geometrically infinite surfaces shows recurrent irregular orbits or non-minimal closures.

We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…

In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow…

The earthquake flow is asymmetric and cannot be extended to an SL(2,R) action.

Study shows orbits on a specific surface without intersecting geodesics.

Proves effective slope gaps for lattice surfaces.

The study connects lamination and orbit closures in hyperbolic manifolds.

Introduces hyperbolic generalized framed surfaces and their properties.

Study saddle connections on hyperelliptic surfaces, finding growth rates.

In this thesis we consider a way to construct a rich family of compact Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with orientation, we construct a finite-area hyperbolic Riemann surface by gluing triangles according to the combinatorics of the graph. We then compactify this surface by adding finit…

A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such $n$-gons are parametrized by the subspaces of $(0,\infty)^n$ that contain their side length collections,…

Extends earthquake and horocycle flows to new measures.

A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.

We study the topological dynamics of the horocycle flow $h_\mathbb{R}$ on a geometrically infinite hyperbolic surface S. Let u be a non-periodic vector for $h_\mathbb{R}$ in T^1 S. Suppose that the half-geodesic $u(\mathbb{R}^+)$ is almost minimizing and that the injectivity radius along $u(\mathbb{R}^+)$ has a finite …

The abstract discusses vector fields on curved spaces and conservation laws.

Classifies actions on complex space forms with Lagrangian orbits.

The study defines and characterizes extrinsic catenaries in hyperbolic space.

Researchers compute gap distributions for saddle connection directions on specific translation surfaces.

We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic measures. These measures are supported on closed immersed manifolds with boundary th…

Convexity properties of Weil-Petersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considera…

We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in S…

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol$_3$. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in tot…

We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to de…

We analyze the slope gap distribution of Veech surfaces, finding finite non-analytic points and quadratic tail decay.

The theory of differential forms began with a discovery of Poincare who found conservation laws of a new type for Hamiltonian systems - The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant differential forms: a symplectic two-form, or a contact one-form for geodesic flo…

Infinite volume moduli spaces of hyperbolic surfaces are redefined with exponential forms.

In this paper we give a complete description of the space $ \QF $ of quasifuchsian punctured torus groups in terms of what we call {\em pleating invariants}. These are natural invariants of the boundary $\bch$ of the convex core of the associated hyperbolic 3-manifold $M$ and give coordinates for the non-Fuchsian group…