arXiv research

The study proves geodesic loops and chords without intersections for specific metrics.

Lower bounds on geodesic length with few intersections on hyperbolic surfaces.

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

Improved bounds on geodesic intersections on hyperbolic surfaces.

Given a hyperbolic surface $§$, a classic result of Birman and Series states that for each $K$, all complete geodesics with at most $K$ self-intersections can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of $§$. We define a self-intersection function for each complete geodesic, which bounds t…

Geodesics with bounded angles have zero Hausdorff dimension.

The paper calculates self-intersections on a pair of pants using Bowen and Series' coding.

The entropy of geodesic currents on hyperbolic surfaces is bounded by their self-intersection number.

On a hyperbolic Riemann surface, given two simple closed geodesics that intersect $n$ times, we address the question of a sharp lower bound $L_n$ on the length attained by the longest of the two geodesics. We show the existence of a surface $S_n$ on which there exists two simple closed geodesics of length $L_n$ interse…

Study on geodesics proving index and intersection bounds, with examples of multiplicity.

Recovering manifold geometry from geodesic intersections.

All closed geodesics are simple and non-intersecting in dimensions 3 and above.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We pr…

The study counts geodesics on curved surfaces with specific intersections.

In this paper we present the algorithms for calculating the differential geometric properties {t,n,b1,b2,b3,k1,k2,k3,k4} along-with geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space R^5. In transversal intersection…

We study closed geodesics on hyperbolic surfaces, and give bounds for their angles of intersection and self-intersection, and for the sides of the polygons that they form, depending only on the lengths of the geodesics

Minimal geodesics on hyperbolic surfaces are long.

Geodesics on hyperbolic surfaces become evenly spread over time.

Paper shows how to evenly distribute intersections in hyperbolic spaces.

New characterization of geodesic currents via curve functionals.

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.

We study the properties of geodesic currents on free groups, particularly the "intersection form" that is similar to Bonahon's notion of the intersection number between geodesic currents on hyperbolic surfaces.

We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting of geodesics. Furthermore, the method gives inequalities with a uniform constant…

Shortest non-simple closed geodesics on hyperbolic surfaces found.

Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.

Paper proves curves can be smoothed to reduce self-intersection by exactly 1.

We investigate the distribution of lengths obtained by intersecting a random geodesic with a geodesic lamination. We give an explicit formula for the distribution for the case of a maximal lamination and show that the distribution is independent of the surface and lamination. We also show how the moments of the distrib…

Constructs geodesics near intersection points of Lagrangian submanifolds.

Study counts geodesic surfaces in knot complements, finding unique ones for small knots.

This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer $k$, we consider the set of closed geodesics that self-intersect at least $k$ times, and investigate those of minimal length. The main result is that, if the surface has at least one cusp, their…

Super efficient geodesics have a unique vertex in the complex of curves.

Study on minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.

Study on complexity of systolic geodesics on Bolza surface.

Study shows nontrivial intersections of subgroups on homogeneous spaces.

Random simple closed curves map Teichmüller space to geodesic currents.

We show that round hemispheres are the only compact 2 dimensional Riemannian manifolds (with or without boundary) such that almost every pair of complete geodesics intersect once and only once. We prove this by establishing a sharp isoperimetric inequality for surfaces with boundary such that every pair of geodesics ha…

Study efficient geodesics in curve complex using dot graphs.

We investigate submanifolds in space forms such that every geodesic orthogonal to the submanifold intersects a fixed totally geodesic submanifold. We obtain an application to horospheres in Hadamard manifolds.

Let $G = \langle A,B \rangle$ be a non-elementary two generator subgroup of the isometry group of $\mathbb{H}^2$, the hyperbolic plane. If $G$ is discrete and free and geometrically finite, its quotient is a pair of pants and in prior work we produced a formula for the number of essential self intersections (ESIs) of a…

Study geodesics of meromorphic connections on Riemann surfaces.

Study shows orbits on a specific surface without intersecting geodesics.

New results on geodesic flows using curve shortening flow.

A pair of distinct free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic is said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of one class is equal to the length of the geodesic representative of the other clas…

We investigate intersections of geodesic lines in $H^2$ and in an associated tree T, proving the following result. Let M be a punctured hyperbolic torus and let $γ$ be a closed geodesic in M. Any edge of any triangle formed by distinct geodesic lines in the preimage of $γ$ in $H^2$ is shorter then $γ$. However, a simil…

Formulae for Masur-Veech volumes and frequencies of geodesics derived from intersection numbers.

The mapping class group of a surface $§$ acts on the set of closed geodesics on $§$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \geq 1$. (The case when $K=0$ is already known.) We also restrict our count to those orbits t…

We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for …

For any geodesic current we associated a quasi-metric space. For a subclass of geodesic currents, called filling, it defines a metric and we study the critical exponent associated to this space. We show that is is equal to the exponential growth rate of the intersection function for closed curves.