Related papers: Lengths of simple loops on surfaces with hyperboli…
We study Thurston's Lipschitz and curve metrics, as well as the arc metric on the Teichmueller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps…
In the Teichm\"uller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to…
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…
In this article, we revisit classical length identities enjoyed by simple closed curves on hyperbolic surfaces. We state and prove the rigidity of such identities over Teichm\"uller spaces. Due to this rigidity, certain collections of…
We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two…
For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy…
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…
We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the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…
There are several Teichm\"uller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint (a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal…
We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite…
Given a measure on the Thurston boundary of Teichmueller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different…
Considering the Teichm\"uller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point…
We study how the length and the twisting parameter of a curve change along a Teichmuller geodesic. We then use our results to provide a formula for the Teichmuller distance between two hyperbolic metrics on a surface, in terms of the…
We develop a natural and geometric way to realize the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces. To do so, a metric is defined on the Teichm\"uller space of the torus, inspired by Thurston's Lipschitz metric…
Compact hyperbolic 3-manifolds are used in cosmological models. Their topology is characterized by their homotopy group $\pi_1(M)$ whose elements multiply by path concatenation. The universal covering of the compact manifold $M$ is the…
We define and study metrics and weak metrics on the Teichmueller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of…
Two free homotopy classes of closed curves in an orientable surface with negative Euler characteristic are said to be length equivalent if for any hyperbolic structure on the surface, the length of the geodesic in one class is equal to the…
We prove that there are Fenchel-Nielsen coordinates for the Teichmueller space of a finite area hyperbolic surface with respect to which the length functions are convex.
We study the action of the elements of the mapping class group of a surface of finite type on the Teichm\"uller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic…
We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient…