Frederick P. Gardiner

Quasiconformal Teichmüller theory

Quasiconformal mapping Riemann surfaces Quadratic differentials, Part I Quadratic differentials, Part II Teichmuller equivalence The Bers embedding Kobayashis metric on Teichmuller space Isomorphisms and automorphisms Teichmuller uniqueness The mapping class group Jenkins-Strebel differentials Measured foliations Obstacle problems Asymptotic Teichmuller space Asymptotically extremal maps Universal Teichmuller space Substantial boundary points Earthquake mappings Bibliography Index.

Extremal length geometry of teichmüller space

Assume τ is a point in the Teichmuller space of a Riemann surface which is compact or obtainable from a compact surface by deleting a finite number of punctures. Let be extermal lengths of two transversely realizable measured folitions on the Riemann surface R r corresponding to the point τ. There is a unique Teichmuller line along which the function is minimum. Teichmuller space embeds into projective classes of vectors of square roots of extremal lengths of simple curves on the base surface. The closure of the image of Teichmuller space under this embedding is compact. Moreover, there is a relationship between the boundary of this embedding and the boundary of the extremal length embedding properly contains the Thruston boundary.

Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces

Let X and Y be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmiiller space of X onto the Teichmiiller space of Y is induced by a quasiconformal homeomorphism of X onto Y. These Teichmiiller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmiiller space and special properties of Teichmiiller disks.

Universal Teichmüller Space

Abstract We present an outline of the theory of universal Teichmtuller space, viewed as part of the theory of QS, the space of quasisymmetric homeomorphisms of a circle. Although elements of QS act in one dimension, most results about QS depend on a two-dimensional proof. QS has a manifold structure modelled on a Banach space, and after factorization by P S L ( 2 , R ) it becomes a complex manifold. In applications, QS is seen to contain many deformation spaces for dynamical systems acting in one, two and three dimensions; it also contains deformation spaces of every hyperbolic Riemann surface, and in this naive sense it is universal. The deformation spaces are complex submanifolds and often have certain universal properties themselves, but those properties are not the object of this article. Instead we focus on the analytic foundations of the theory necessary for applications to dynamical systems and rigidity.

Infinitesimal bending and twisting in one-dimensional dynamics

An infinitesimal theory for bending and earthquaking in one- dimensional dynamics is developed. It is shown that any tangent vector to Teichmuller space is the initial data for a bending and for an earthquaking or- dinary differential equation. The discussion involves an analysis of infinitesimal bendings and earthquakes, the Hilbert transform, natural bounded linear oper- ators from a Banach space of measures on the Mobius strip to tangent vectors to Teichmuller space, and the construction of a nonlinear right inverse for these operators. The inverse is constructed by establishing an infinitesimal version of Thurstons earthquake theorem.

Comparing Poincare densities

However, there is no formula for the Poincare density pn of a general plane domain Q, and so one must resort to estimates. Many research articles have been devoted to finding and working with such estimates (see for example [4], [5], [15], [19], [24], and [25]). The first tool to establish such estimates is Schwarzs lemma which says that every holomorphic map is a contraction with respect to the Poincare metric. That is, ps(f)ldf I< pD(z)Idzl, for any holomorphic map f from a Riemann surface D to another Riemann surface Q. Applying this lemma

A hyperelliptic realization of the horseshoe and baker maps

We present a generalization of the functional equation for the Weierstrassk

Non-special divisors supported on the branch set of a p-gonal Riemann surface

\wp

A theorem of Bers and Greenberg for infinite dimensional Teichmüller spaces

-function for hyperelliptic surfaces of infinite genus arising from iteration of the horseshoe and baker maps. The ramified cover of these infinite genus surfaces over the complex plane are associated to a quadratic differential of finite norm with simple poles accumulating to infinity. We study the geometry of its critical trajectories emanating from these poles and their rate of accumulation.

On completing triangles in infinite dimensional teichmüller spaces

A compact Riemann surface S is called cyclic p-gonal if it possesses an automorphism τ of order p such that the quotient S/ < τ > has genus zero. It is well known that if p is a prime number and Q1, . . . , Qr ∈ S are the fixed points of τ then S has genus g = p−1 2 (r − 2). In this article we find a criterion to decide when a divisor of the form D = Q1 1 · · ·Qdr r , with ∑ di = g, is non-special. The criterion is very easy to apply in practice since it only depends on the arithmetic of the local rotation numbers of τ at the points Qi and the multiplicities of these points on the divisor D, i.e. the integers di. Knowledge of the set of non-special divisors supported on the ramification set seems to be essential in all attempts to extend the classical Thomae formulae, which apply to hyperelliptic (i.e. 2-gonal) Riemann surfaces, to the case of p-gonal ones. Notation. Throughout this paper we use the following notation. Given an integer n ∈ Z we shall denote by n ∈ {0, 1, ..., p − 1} and [n] ∈ Z/pZ its remainder and its residue class modulo p, respectively; thus, we have [n] = [n]. 1.