Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces

Let g : M -> N be a quasiconformal harmonic diffeomorphism between noncompact Riemann surfaces M and N. In this paper we study the relation between the map g and the complex structures given on M and N. In the case when M and N are of finite analytic type we derive a precise estimate which relates the map g and the Teichmüller distance between complex structures given on M and N. As a corollary we derive a result that every two quasiconformally related finitely generated Kleinian groups are also related by a harmonic diffeomorphism. In addition, we study the question of whether every quasisymmetric selfmap of the unit circle has a quasiconformal harmonic extension to the unit disk. We give a partial answer to this problem. We show the existence of the harmonic quasiconformal extensions for a large class of quasisymmetric maps. In particular it is proved that all symmetric selfmaps of the unit circle have a unique quasiconformal harmonic extension to the unit disk

Convex regions in the plane and their domes

We make a detailed study of the relation of a euclidean convex region $\Omega \subset \mathbb C$ to $\mathrm{Dome} (\Omega)$. The dome is the relative boundary, in the upper halfspace model of hyperbolic space, of the hyperbolic convex hull of the complement of $\Omega$. The first result is to prove that the nearest point retraction $r: \Omega \to \mathrm{Dome} (\Omega)$ is 2-quasiconformal. The second is to establish precise estimates of the distortion of $r$ near $\partial \Omega$

The Teichmüller distance between finite index subgroups of PSL2(Z)

For a given 0 , we show that there exist two finite index subgroups of PSL2(Z) which are (1+) -quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any 0 there are two finite regular covers of the Modular once punctured torus T 0 (or just the Modular torus) and a (1+) -quasiconformal map between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S p ) of the punctured solenoid S p under the action of the corresponding Modular group (which is the mapping class group of S p [6], [7]) has the closure in T(S p ) strictly larger than the orbit and that the closure is necessarily uncountable

The universal properties of Teichmueller spaces

We discuss universal properties of general Teichmüller spaces. Our topics include the Teichmüller metric and Kobayashi metric, extremality and unique extremality of quasiconformal mappings, biholomorphic maps between Teichmüller space, earthquakes and Thurston boundary

Immersing almost geodesic surfaces in a closed hyperbolic three manifold

Let M3 be a closed hyperbolic three manifold. We construct closed surfaces that map by immersions into M3 so that for each, one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection

Quasiconformal homogeneity of genus zero surfaces

A Riemann surface M is said to be K-quasiconformally homogeneous if, for every two points p, q ∈ M, there exists a K-quasiconformal homeomorphism f : M→M such that f (p) = q. In this paper, we show there exists a universal constant K > 1 such that if M is a K-quasiconformally homogeneous hyperbolic genus zero surface other than D2, then K ≥ K. This answers a question by Gehring and Palka [10]. Further, we show that a non-maximal hyperbolic surface of genus g ≥ 1 is not K-quasiconformally homogeneous for any finite K ≥ 1

The logarithmic spiral : a counterexample to the K=2 conjecture

Given a nonempty compact connected subset X subset of S-2 with complement a simply-connected open subset Omega subset of S-2, let Dome (Omega) be the boundary of the hyperbolic convex hull in H-3 of X. We show that if X is a certain logarithmic spiral, then we obtain a counterexample to the conjecture of Thurston and Sullivan that there is a 2-quasiconformal homeomorphism Omega -> Dome (Omega) which extends to the identity map on their common boundary in S-2. This leads to related counterexamples when the boundary is real analytic, or a finite union of intervals (straight intervals, if we take S-2 = C boolean OR {infinity}). We also show how this counterexample enables us to construct a related counterexample which is a domain of discontinuity of a torsion-free quasifuchsian group with compact quotient. Another result is that the average long range bending of the convex hull boundary associated to a certain logarithmic spiral is approximately .98 pi/2, which is substantially larger than that of any previously known example

Complex earthquakes and deformations of the unit disk

We define deformations of certain geometric objects in hyperbolic 3-space. Such an object starts life as a hyperbolic plane with a measured geometric lamination. Initially the hyperbolic plane is embedded as a standard hyperbolic subspace. Given a complex number t, we obtain a corresponding object in hyperbolic 3-space by earthquaking along the lamination, parametrized by the real part of t, and then bending along the image lamination, parametrized by the complex part of t. In the literature, it is usually assumed that there is a quasifuchsian group that preserves the structure, but this paper is more general and makes no such assumption. Our deformation is holomorphic, as in the lambda-lemma, which is a result that underlies the results in this paper. Our deformation is used to produce a new, more natural proof of Sullivan's theorem: that, under standard topological hypotheses, the boundary of the convex hull in hyperbolic 3-space of the complement of an open subset U of the 2-sphere is quasi-conformally equivalent to U, and that, furthermore, the constant of quasiconformality is a universal constant. Our paper presents a precise statement of Sullivan's Theorem. We also generalize much of McMullen's Disk Theorem, describing certain aspects of the parameter space for certain parametrized spaces of 2-dimensional hyperbolic structures