Irwin Kra

On Teichmüller’s theorem on the quasi-invariance of cross ratios

Teichmüller’s theorem gives necessary and sufficient conditions for mapping one ordered quadruple by aK-quasiconformal map onto a second ordered quadruple. We give a simple non-computational proof of the necessity part. We then characterize such extremal mappings, and obtain as a consequence a new formula for the modular function, with leads to a very simple derivation of the known expression for the Poincaré metric on the thrice-punctured sphere.

AUTOMORPHIC FORMS FOR SUBGROUPS OF THE MODULAR GROUP

The theory of theta constants with rational characteristics is developed from the point of view of automorphic functions for the principal congruence subgroups of the modular group PSL(2, ℤ). New identities are derived and particular emphasis is given to the level 3 case where a striking generalization of the classical λ-function is obtained.

Families of univalent functions and Kleinian groups

The paper treats two topics. We obtain new characterizations of quasidiscs using theλ-lemma. We study regularb-groups by combining results on holomorphic families of injections with theorems on algebraic convergence of Kleinian groups and the structure of the Eichler cohomology spaces. We also obtain results about projective structures whose developing maps are coverings.

Branched two-sheeted covers

In this paper we explore the connection between Weierstrass points of subspaces of the holomorphic differentials and the geometry of the canonical curve inPCg−1. In particular, we consider non-hyperelliptic Riemann surfaces with involution and the Weierstrass points of the −1 eigenspace of the holomorphic differentials. The case of coverings of a torus is considered in detail.

Sequences and Series of Holomorphic Functions

We now turn from the study of a single holomorphic function to the investigation of collections of holomorphic functions. In the first section we will see that under the appropriate notion of convergence of a sequence of holomorphic functions, the limit function inherits several properties that the approximating functions have, such as being holomorphic, and in the second section, we show that the space of holomorphic functions on a domain can be given the structure of a complete metric space. We then apply these ideas to obtain a series expansion for the cotangent function.

The Fundamental Theorem in Complex Function Theory

This introductory chapter is meant to convey the need for and the intrinsic beauty found in passing from a real variable x to a complex variable z. In the first section we “solve” two natural problems using complex analysis. In the second, we state what we regard as the most important result in the theory of functions of one complex variable, which we label the fundamental theorem of complex function theory, in a form suggested by the teaching and exposition style of Lipman Bers; its proof will occupy most of this volume. The next two sections of this chapter include an outline of our plan for the proof and an outline for the text, respectively; in subsequent chapters we will define all the concepts encountered in the statement of the theorem in this chapter. The reader may not be able at this point to understand all (or any) of the statements in our fundamental theorem or to appreciate its depth and might choose initially to skim this material. All readers should periodically, throughout their journey through this book, return to this chapter, particularly to the last section, that contains an interesting account of part of the history of the subject.

The Cauchy Theory: Key Consequences

This chapter is devoted to some immediate consequences of the Fundamental Theorem for Cauchy Theory, Theorem 4.52, of the last chapter. Although the chapter is very short, it includes proofs of many of the implications of the Fundamental Theorem 1.1. We point out that these relatively compact proofs of a host of major theorems result from the work put in Chapter 4 and earlier chapters.

Conformal Equivalence and Hyperbolic Geometry

In this chapter, we study conformal maps between domains in the extended complex plane \(\widehat{\mathbb{C}}\); these are one-to-one meromorphic functions. Our goal here is to characterize all simply connected domains in the extended complex plane. The first two sections of this chapter study the action of a quotient of the group of two-by-two nonsingular complex matrices on the extended complex plane, namely, the group PSL(2, ℂ) and the projective special linear group. This group is also known as the Mobius group. In the third section we characterize simply connected proper domains in the complex plane by establishing the Riemann mapping theorem (RMT). This extraordi- nary theorem tells us that there are conformal maps between any two such domains.

Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions

In this chapter we use the Cauchy Theory to study functions that are holomorphic on an annulus and analytic functions with isolated singularities. We describe a classification for isolated singularities. Functions that are holomorphic on an annulus have Laurent series expansions, an analog of power series expansions for holomorphic functions on disks. Holomorphic functions with a finite number of isolated singularities in a domain can be integrated using the Residue Theorem, an analog of the Cauchy Integral Formula. We discuss the local properties of these functions.

In Ramanujan’s Shadow

One variable θ-functions and θ-constants are powerful tools in the study of function theory, combinatorics and number theory. This is the main theme of this article and our recently published book [2]. We emphasize a unified approach to a diverse set of problems that include uniformizations of Riemann surfaces represented by subgroups of the modular group, applications to combinatorial number theory and the derivation of beautiful identities. In this expository note we provide a sample of the methods and results of my joint work with H.M Farkas. We outline some proofs and supply complete references to results not appearing in [2].