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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