Preface

Lars Ahlfors’s book Lectures on Quasiconformal Mappings was first published

in 1966, and its special qualities were soon recognized. For example, a Russian

translation was published in 1969, and, after seeing an early version of the notes that

were the basis for Ahlfors’s book, Lipman Bers, Fred Gardiner and Kra abandoned

their plans to produce a book based on Bers’s two-semester 1964 course at Columbia

on quasiconformal mappings and Teichm¨ uller spaces.

Ahlfors’s classic continues to be widely read by graduate students and other

mathematicians who are learning the foundations of the theories of quasiconfor-

mal mappings and Teichm¨ uller spaces. It is particularly suitable for that purpose

because of the elegance with which it presents the fundamentals of the theory of

quasiconformal mappings. The early chapters provide precisely what is needed for

the big results in Chapters V and VI. At the same time they give the reader an

informative picture of how quasiconformal mappings work.

One reason for the economy of Ahlfors’s presentation is that his book represents

the contents of a one-semester course, given at Harvard University in the spring

term of 1964. It was a remarkable achievement; in one semester he developed the

theory of quasiconformal mappings from scratch, gave a self-contained treatment

of Beltrami’s equation (Chapter V of the book), and covered the basic properties

of Teichm¨ uller space, including the Bers embedding and the Teichm¨ uller curve

(see Chapter VI and §2 of our chapter in the appendix). Along the way, Ahlfors

found time for some estimates in Chapter III B involving elliptic integrals and a

treatment of an extremal problem of Teichm¨ uller in Chapter III D that even now

can be found in few other sources. The fact that quasiconformal mappings turned

out to be important tools in 2 and 3-dimensional geometry, complex dynamics and

value distribution theory created a new audience for a book that provides a uniquely

eﬃcient introduction to the subject. It illustrates Ahlfors’s remarkable ability to

get straight to the heart of the matter and present major results with a minimum

set of prerequisites.

The notes on which the book is based were written by Ahlfors himself. It was

his practice in advanced courses to write thorough lecture notes (in longhand, with

a fountain pen), leaving them after class in a ring binder in the mathematics library

reading room for the benefit of the people attending the course.

With this practice in mind, Fred Gehring invited Ahlfors to publish the spring

1964 lecture notes in the new paperback book series Van Nostrand Mathematical

Studies that he and Paul Halmos were editing. Ahlfors, in turn, invited his recent

student Earle, who had completed his graduate studies and left Harvard shortly

before 1964, to edit the longhand notes and see to their typing. The published text

hews close to the original notes, and of course Ahlfors checked and approved the

few alterations that were suggested.

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