Related papers: Carath\'eodory metric on some generalized Teichm\"…
We establish a simple and powerful lemma that provides a criterion for sequences in metric spaces to be Cauchy. Using the lemma, it is then easily verified that the Picard iterates $\{T^nx\}$, where $T$ is a contraction or asymptotic…
We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a…
This paper shows that every totally-geodesic isometry from the unit disk to a finite-dimensional Teichm\"uller space for the intrinsic Kobayashi metric is either holomorphic or anti-holomorphic; in particular, it is a Teichm\"uller disk.…
Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a…
We show that the horofunction boundary of Teichm\"uller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases.…
According to Sullivan, a space ${\cal E}$ of unimodal maps with the same combinatorics (modulo smooth conjugacy) should be treated as an infinitely-dimensional Teichm\"{u}ller space. This is a basic idea in Sullivan's approach to 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…
Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ with the Teichm\"uller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration…
Thurston introduced in his seminal work an asymmetric metric on Teichm\"uller space by the ratio of simple closed curve length. In this paper, we generalize the idea and define an asymmetric metric on the space of unit-area flat metrics…
Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and…
We prove a semisimplicity result for the boundary, in the corresponding Deligne-Mumford compactification, of a totally geodesic subvariety of a moduli space of Riemann surfaces. At the level of Teichm\"uller space, this semisimplicity…
We lay foundations of the subject in the title, on which we build in another paper devoted to isometries in spaces of K\"ahler metrics.
We show how classical differentiation theorems for measures can be turned into an integral representation of a Borel measure with respect to a fixed Carath\'eodory measure. We focus our attention on the cases where this measure is both the…
We study the Bers isomorphism between the Teichm\"uller space of the parabolic cyclic group and the universal Teichm\"uller curve. We prove that this is a group isomorphism and its derivative map gives a remarkable relation between Fourier…
The Universal Teichm\"uller Space, $T(1)$, is a universal parameter space for all Riemann surfaces. In earlier work of the first author it was shown that one can canonically associate infinite- dimensional period matrices to the coadjoint…
Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show that the natural action of the group of volume-preserving…
Teichm\"uller curves play an important role in the study of dynamics in polygonal billiards. In this article, we provide a criterion similar to the original M\"oller's criterion, to detect whether a complex curve, embedded in the moduli…
This is an introduction to the algebraic aspect of Teichm\"uller dynamics, with a focus on its interplay with the geometry of moduli spaces of curves as well as recent advances in the field.
In this paper, we study the asymptotic geometry of Teichmuller space of Riemann surfaces and give bounds on the Weil-Petersson sectional curvature of Teichmuller space, in terms of the length of the shortest geodesic on the surface. This…
We show that on a certain class of bounded, complete Reinhardt domains in $\mathbb{C}^n$ that enjoy a lot of symmetries, the Carath\'eodory pseudo-distance and the geodesic distance of the complete K\"ahler-Einstein metric with Ricci…