Lectures on Quasiconformal Mappings

by Lars Valerian Ahlfors

Lars Ahlfors' Lectures on Quasiconformal Mappings, based on a course he gave at Harvard University in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. These lectures develop the theory of quasiconformal mappings from scratch, give a self-contained treatment of the Beltrami equation, and cover the basic properties of Teichmuller spaces, including the Bers embedding and the Teichmuller curve. It is remarkable how Ahlfors goes straight to the heart of the matter, presenting major results with a minimum set of prerequisites. Many graduate students and other mathematicians have learned the foundations of the theories of quasiconformal mappings and Teichmuller spaces from these lecture notes. This edition includes three new chapters.The first, written by Earle and Kra, describes further developments in the theory of Teichmuller spaces and provides many references to the vast literature on Teichmuller spaces and quasiconformal mappings. The second, by Shishikura, describes how quasiconformal mappings have revitalized the subject of complex dynamics. The third, by Hubbard, illustrates the role of these mappings in Thurston's theory of hyperbolic structures on 3-manifolds. Together, these three new chapters exhibit the continuing vitality and importance of the theory of quasiconformal mappings.

Genres: Mathematics

162 pages, Paperback

First published May 30, 1987

About the author

Lars Valerian Ahlfors, William Caspar Graustein Professor of Mathematics, died of pneumonia on Oct. 11 in Pittsfield, Mass., at the age of 89. Ahlfors won the first Fields Medal awarded by the International Mathematics Society in 1936, a quadrennial award considered equivalent, in mathematics, to the Nobel Prize. In 1979, he was awarded the prestigious Wolf Prize in Jerusalem. Ahlfors was best known for his work in complex analysis, a fundamental subject with many applications from number theory to modern physics. His textbook, Complex Analysis, first published in 1953, with new editions in 1966 and 1979, is still considered the leading text in the field. He wrote three other mathematical books and published almost 100 papers. Colleagues and students have described his work as extraordinarily elegant, and his lectures, delivered in thundering basso, as stunningly beautiful. Ahlfors was born in Helsinki, Finland, in 1907. As he often observed, his mother's death during his birth critically influenced his life. His father, a professor of mechanical engineering at the Polytechnic Institute, was attentive but stern. In the brief autobiographical note that introduced his collected papers, published in 1982, he wrote that, "As a child, I was fascinated by mathematics without understanding what it was about, but I was by no means a child prodigy. As a matter of fact, I had no access to any mathematical literature except in the highest grades. . . . The high school curriculum did not include any calculus but I finally managed to learn some on my own, thanks to clandestine visits to my father's engineering library." Ahlfors' introduction to higher mathematics, including complex analysis, came from his mentors at Helsinki University, Ernst Lindelof and Rolf Nevanlinna. At 21, Ahlfors followed Nevanlinna to the Federal Polytechnic Institute in Zurich, where he began working at a research level. There he produced his first major work, a study of asymptotic values of an entire function, based on his own new approach to conformal mapping. Self-effacingly, Ahlfors credited Nevanlinna and another teacher, George Polya, for their "considerable help." They, in turn, insisted that he publish the results solely in his own name. Thereafter, as he expressed it, "I have tried to repay my debt by never accepting to appear as coauthor with a student."