Related papers: Trois applications du lemme de Schwarz aux surface…

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple…

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths…

We obtain geometric lower bounds for the low Steklov eigenvalues of finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain depend on the length of a shortest multi-geodesic disconnecting the surfaces into connected…

We study the injectivity radius of complete Riemannian surfaces (S,g) with curvature |K(g)| bounded by 1. We show that if S is orientable with nonabelian fundamental group, then there is a point p in S with injectivity radius at least…

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 prove that, if a closed geodesic $\Gamma$ on a complete finite type hyperbolic surface has at least 2 self-intersections, then the length of $\Gamma$ has an lower bound $2\log(5+2\sqrt6)$, and the lower bound is sharp, attained on a…

Let $h:X \to Y$ be a homeomorphism between hyperbolic surfaces with finite topology. If $h$ is homotopic to a holomorphic map, then every closed geodesic in $X$ is at least as long as the corresponding geodesic in $Y$, by the Schwarz Lemma.…

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.

We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a \lq\lq stable neighborhood\rq\rq whose width only…

The Schwarzian derivative provides a classical analytic measure of how far a holomorphic map of the disk is from being M\"obius, with Nehari's bounds giving sharp criteria for univalence. Independently, Thurston introduced a geometric…

In 1991, Beardon and Stephenson [2] generalized the classical Schwarz-Pick lemma in hyperbolic geometry to the discrete Schwarz-Pick lemma for Andreev circle packings. This paper continues to investigate the discrete Schwarz-Pick lemma for…

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

On a closed Riemannian surface of negative curvature, we prove a characterization for configurations of closed geodesics arising from one parameter Allen-Cahn min-max constructions. One of the facts we conclude is that every geodesic occurs…

We describe the "hyperbolic" properties of a riemann surface lamination M canonically associated to every compact three manifolds of curvature less than 1. More precisely, if the geodesic flow is the phase space attached to an ordinary…

In this paper we obtain a bound on the number of isometry classes of finite area hyperbolic surfaces which are length isospectral to a given surface depending only on the topological type of the surface and the length of the shortest closed…

In this paper, maximum principles for Euclidean and hyperbolic discrete conformal structures on polyhedral surfaces are established. These maximum principles unify and generalize the maximum principles for vertex scalings and different…

We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.

Given a compact orientable surface with finitely many punctures $\Sigma$, let $\Cal S(\Sigma)$ be the set of isotopy classes of essential unoriented simple closed curves in $\Sigma$. We determine a complete set of relations for a function…

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…

We study the order of lengths of closed geodesics on hyperbolic surfaces. Our first main result is that the order of lengths of curves determine a point in Teichm\"uller space. In an opposite direction, we identify classes of curves whose…