Clifford J. Earle

Isometrically embedded polydisks in infinite dimensional Teichmüller spaces

We define isometric holomorphic embeddings of the infinite dimensional polydisk D∞ in any infinite dimensional Teichmüller space. These embeddings provide simple new proofs that the Teichmüller metric on any infinite dimensional Teichmüller space is non-differentiable and has arbitrarily short simple closed geodesics. They also lead to a complete characterization of the points in Teichmüller space that lie on more than one straight line through the basepoint.

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.

Quasiconformal variation of slit domains

We use quasiconformal variations to study Riemann mappings onto variable single slit domains when the slit is the tail of an appropriately smooth Jordan arc. In the real analytic case our results answer a question of Dieter Gaier and show that the function κ in Lowners differential equation is real analytic.

Conformally natural reflections in Jordan curves with applications to Teichmüller spaces

In his fundamental paper [1] Ahlfors initiated the study of quasiconformal reflections. Using the results of Beurling and Ahlfors [6] he showed that every quasicircle that passes through ∞ permits a quasiconformal reflection that satisfies a global Lipschitz condition (with exponent one) in the plane. Using that result he proved by a direct construction that the Bers embedding of the universal Teichmuller space has an open image. Lipschitz continuous quasiconformal reflections also play a crucial role in Bers’s subsequent proof (see [4] and [5]) that for any Teichmuller space the Bers embedding not only has an open image but also has local cross sections. That result is one of the cornerstones of Teichmuller theory.

Extremal Quasiconformal Mappings in Plane Domains

This paper is an expanded version of the lecture given by the first author at the Ann Arbor conference celebrating Fred Gehring’s seventieth birthday and his many contributions to mathematics. Our primary purpose is to give an exposition of some of the main results of our joint paper [EL]. In view of the special occasion, we have devoted Sections 2 through 8 to a survey of some classical results about extremal quasiconformal mappings Sections 9 and 10 report some results from [EL], and Section 11 reports some results obtained by Nikola Lakic while this paper was in progress. The interested reader can learn more about extremal mappings from Kurt Strebel’s survey article [St3] and Edgar Reich’s paper [R3] in this volume, both of which we highly recommend.

Angular derivatives of the barycentric extension

Let S 1 be the boundary of the open unit disk D and let be the barycentric extension of the sense-preserving homeomorphism . We prove that has an angular derivative at any boundary point where has a non-zero derivative. The proof depends on characterizing certain differentiability properties of and in terms of their behavior under composition with certain conformal automorphisms of D.

On Quasiconformal Extensions of the Beurling–Ahlfors Type

Publisher Summary Beurling and Ahlfors had determined which homeomorphisms of the real line are the boundary values of quasiconformal self-maps of the upper half-plane. Further, for each such homeomorphism of the real line, they found an explicit quasiconformal extension. Later, the clifford J. Earle and Eells showed that the Beurling–Ahlfors extension depends continuously on the boundary values and, therefore, provides a continuous cross section of the universal Teichmuller space. This cross section is real analytic. The chapter presents a theorem, which gives a family of real analytic cross sections depending holomorphically on a complex parameter τ. It also presents another theorem, which applies the earlier results to the Bers fiber space over universal Teichmuller space.

The Integrable automorphic forms as a dual space, II

We show that the Banach space of integrable automorphic forms on any bounded domain B in is isometrically isomorphic to the dual space of a certain subspace of its own dual space.

A Montel Theorem for Holomorphic Functions on Infinite Dimensional Spaces that Omit the Values 0 and 1

Let X be a connected complex Banach manifold. We give a proof that the set of holomorphic functions on X that omit the values 0 and 1 is a normal family.