Related papers: Ordering curves on surfaces

There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…

The present paper deals with a study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that…

Let $\Sigma$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $\Sigma$. A geodesic curve $\gamma$ in $\Sigma$ is filling, if it cuts the surface into topological disks and…

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…

We provide a combinatorial condition characterizing curves that are short along a Teichmueller geodesic. This condition is closely related to the condition provided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that…

Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two…

We are concerned with mapping class groups of surfaces with nonempty boundary. We present a very natural method, due to Thurston, of finding many different left orderings of such groups. The construction involves equipping the surface with…

Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus $g$ for which any pants decomposition requires curves of total length at least…

This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other…

Suppose that N is a geometrically finite orientable hyperbolic 3-manifold. Let P(N,C) be the space of all geometrically finite hyperbolic structures on N whose convex core is bent along a set C of simple closed curves. We prove that the map…

This article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality…

The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the…

We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of…

A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first…

Through the Schwarz lemma, we provide a new point of view on three well-known results of the geometry of hyperbolic surfaces. The first result deal with the length of closed geodesics on hyperbolic surfaces with boundary (Thurston, Parlier,…

Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[\gamma]$ of a closed curve $\gamma \in \pi_{1}(S_{g,n})$ contains a unique closed…

We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichm\"uller theory than arbitrary non-compact surfaces. We show that the…

We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is…

A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $\Pi$ is considered. A problem of determination of such surfaces when both principal curvatures are given…

The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $g$ of a fixed arithmetic surface…