Related papers: Shortest k-Geodesics on Hyperbolic Surfaces
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…
The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular $k$-gonal tile…
Let $(M, \partial M)$ be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that the boundary is smooth and strictly convex. We show that the induced…
Dumas, Foucaud, Perez, and Todinca [SIAM J. Disc. Math., 2024] proved that if the vertex set of a graph $G$ can be covered by $k$ shortest paths, then the pathwidth of $G$ is bounded by $\mathcal{O}(k \cdot 3^k)$. We prove a coarse variant…
This paper introduces a combinatorial structure of orthogeodesics on hyperbolic surfaces and presents several relations among them. As a primary application, we propose a recursive method for computing the trace (the hyperbolic cosine of…
For any given natural number $k$, this paper gives upper bounds on the radius of a packing of a complete hyperbolic surface of finite area by $k$ equal-radius disks in terms of the surface's topology. We show that the bounds given here are…
Given a topological cell decomposition of a closed surface equipped with edge weights, we consider the Dirichlet energy of any geodesic realization of the 1-skeleton graph to a hyperbolic surface. By minimizing the energy over all possible…
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
Let $M$ be a simply connected Riemannian manifold in $\mathscr{M}_{k,v}^D(n)$, the space of closed Riemannian manifolds of dimension $n$ with sectional curvature bounded below by $k$, volume bounded below by $v$, and diameter bounded above…
In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we show that a geodesically complete hyperbolic surface is made up of its convex…
We examine closed geodesics in the quotient of hyperbolic three space by the discrete group of isometries SL(2,Z[i]). There is a correspondence between closed geodesics in the manifold, the complex continued fractions originally studied by…
Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and…
We study the \emph{geometric $k$-colored crossing number} of complete graphs $\overline{\overline{\text{cr}}}_k(K_n)$, which is the smallest number of monochromatic crossings in any $k$-edge colored straight-line drawing of $K_n$. We…
The hexagon is the least-perimeter tile in the Euclidean plane. On hyperbolic surfaces, the isoperimetric problem differs for every given area. Cox conjectured that a regular $k$-gonal tile with 120-degree angles is isoperimetric for its…
In the presence of certain topological conditions, we provide lower bounds for the infimum of the length function associated to a collection of curves on Teichm\"uller space that depend on the dual cube complex associated to the collection,…
The systoles of a hyperbolic surface {\Sigma} are the shortest closed geodesics. We say that the systoles fill the surface if the set Syst({\Sigma}) of all systoles cuts {\Sigma} into polygons. We refine an idea of Schmutz [15] to construct…
Let $G$ be a relatively hyperbolic group that admits a decomposition into a finite graph of relatively hyperbolic groups structure with quasi-isometrically (qi) embedded condition. We prove that the set of conjugates of all the vertex and…
Finding a totally geodesic surface, an embedded surface where the geodesics in the surface are also geodesics in the surrounding manifold, has been a problem of interest in the study of 3-manifolds. This has especially been of interest in…
In this work, we study the geodesics of the space of certain geometrically and physically motivated subspaces of the space of immersed curves endowed with a first order Sobolev metric. This includes elastic curves and also an extension of…