Related papers: Vertical Limits of Graph Domains
We prove a quantitative version of the non-uniform hyperbolicity of the Teichm\"uller geodesic flow. Namely, at each point of any Teichm\"uller flow line, we bound the infinitesimal spectral gap for variations of the Hodge norm along the…
In this paper, we discuss the boundary behavior of bounded pluriharmonic functions on the Teichm\"uller space. We will show a version of the Fatou theorem that every bounded pluriharmonic function admits the radial limits along the…
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…
We construct an example of a quadratic differential whose vertical foliation is uniquely ergodic and such that the Teichmuller geodesic determined by the quadratic differential diverges in the moduli space of Riemann surfaces.
We propose the graph description of Teichm\"uller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe…
We show that the Teichm\"uller space of a surface without boundary and with punctures, equipped with Thurston's metric is the limit (in an appropriate sense) of Teichm\"uller spaces of surfaces with boundary, equipped with their arc…
We study the geometry of the Thurston metric on the Teichm\"uller space $\mathcal{T}(S)$ of hyperbolic structures on a surface $S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type;…
We construct an example of a Teichmueller geodesic ray whose limit set in Thurston boundary of Teichmueller space is a d-dimensional simplex.
We consider a quotient space of the Bers boundary of Teichm\"{u}ller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space into a point. This reduced Bers boundary turns out to be independent…
The Teichm\"{u}ller curve is the fiber space over Teichm\"{u}ller space of closed Riemann surfaces, where the fiber over a point in Teichm\"{u}ller space is the underlying surface. We derive formulas for sectional curvatures on the…
Let $\varphi :\mathbb{D}\to\mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_{\varphi}:\mathbb{D}\to T(\mathbb{D})$ the Teichm\"uller disk in the universal Teichm\"uller space $T(\mathbb{D})$. For a…
We study the degeneration of hyperbolic surfaces along a ray given by the harmonic map parametrization of Teichm\"uller space. The direction of the ray is determined by a holomorphic quadratic differential on a punctured Riemann surface,…
A basic feature of Teichm\"uller theory of Riemann surfaces is the interplay of two dimensional hyperbolic geometry, the behavior of geodesic-length functions and Weil-Petersson geometry. Let $\mathcal{T}_g$ $(g\geq 2)$ be the Teichm\"uller…
Teichm\"uller's classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself, and maps a given…
We give two flexible and degenerate constructions related to a theorem of Thurston. First, we produce geodesic segments for Thurston's asymmetric metric on Teichm\"uller space $\mathcal{T}(S_g)$ that remain geodesics after adding arbitrary…
In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichm\"uller space. On the other hand, we construct examples of…
We outline old and new results concerning the well-known problems in the Teichm\"uller space theory, i.e., whether these spaces are starlike in the Bers holomorphic embedding and whether any Teichm\"uller space of dimension greater than 1…
In the moduli space of quadratic differentials over complex structures on a surface, we construct a set of full Hausdorff dimension of points with bounded Teichm\"uller geodesic trajectories.The main tool is quantitative nondivergence 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…
We prove that the Teichm\"{u}ller space $\mathscr{T}$ of a closed surface of genus $g \ge 2$ cannot be biholomorphic to any domain which is locally strictly convex at some boundary point.