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We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph…

We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are…

We study the Teichm\"uller metric on the Teichm\"uller space of a surface of finite type, in regions where the injectivity radius of the surface is small. The main result is that in such regions the Teichm\"uller metric is approximated up…

On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces…

In the 1970s, the collar theorem was proven, establishing the existence of uniform tubular neighborhoods of simple closed geodesics on compact surfaces, whose widths depend only on the lengths of the geodesics and the lower bound of the…

Unlike the case of surfaces of topologically finite type, there are several different Teichm\"uller spaces that are associated to a surface of topological infinite type. These Teichm\"uller spaces first depend (set-theoretically) on whether…

In this paper, the Teichm{\"u}ller spaces of surfaces appear from two points of views: the conformal category and the hyperbolic category. In contrast to the case of surfaces of topologically finite type, the Teichm{\"u}ller spaces…

Let $X_{0}$ be a complete hyperbolic surface of infinite type with geodesic boundary which admits a countable pair of pants decomposition. As an application of the Basmajian identity for complete bordered hyperbolic surfaces of infinite…

We construct an example of a Teichmueller geodesic ray whose limit set in Thurston boundary of Teichmueller space is a d-dimensional simplex.

A piecewise flat Finsler metric on a triangulated surface $M$ is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of…

If $p : Y \to X$ is an unramified covering map between two compact oriented surfaces of genus at least two, then it is proved that the embedding map, corresponding to $p$, from the Teichm\"uller space ${\cal T}(X)$, for $X$, to ${\cal…

Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichm\"uller spaces, and hyperbolic 3-manifolds. In the same paper, they…

In an infinite-dimensional Teichm\"uller space, it is known that the geodesic connecting two points can be unique or not. In this paper, we study the situation on the geodesic in the universal asymptotic Teichm\"uller space $AT(\Delta)$. We…

We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have…

The present paper is devoted to the investigation of absolutely continuous curves in asymmetric metric spaces induced by Finsler structures. Firstly, for asymmetric spaces induced by Finsler manifolds, we show that three different kinds of…

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 give explicit bounds on the intersection number between any curve on a tight multigeodesic and the two ending curves. We use this to construct all tight multigeodesics and so conclude that distances in the curve graph are computable. The…

In this paper, we introduce a new asymmetric weak metric on the Teichm{\"u}ller space of a closed orientable surface with (possibly empty) punctures.This new metric, which we call the Teichm{\"u}ller-Randers metric, is an asymmetric…

For a finitely generated group $G$, we introduce an asymmetric pseudometric on projectivized deformation spaces of $G$-trees, using stretching factors of $G$-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space…

Over the past two decades the theory of the Weil-Petersson metric has been extended to general Teichm\"uller spaces of infinite type, including for example the universal Teichm\"uller space. In this paper we give a survey of the main…