extensions of Morrey's classical result [87] on the existence of homeomorphic
solutions to the Beltrami equation, what has come to be known as the "measur-
able Riemann mapping theorem" and extend more recent work of David [33] and
Brakolova-Jenkins [26] which we discuss below.
Quasiconformal mappings (the homeomorphic solutions of Beltrami equations)
provide a class of mappings which lie "between" homeomorphisms and diffeomor-
phisms and yet retain many of the features of analytic functions. They were first
used in a significant way by Ahlfors in 1935 [1] when they proved to be an essen-
tial ingredient of his geometric development of Nevanlinna theory via the theory of
covering spaces and Riemann surfaces.
Using the Beltrami equation, Bers [19] gave a careful exposition of the method
of quasiconformal mappings applied to the classical problems of uniformization in
complex analysis. Bers gave a proof that every homotopy class contains a locally
quasiconformal homeomorphism of one Riemann surface onto another and he also
gave complete and independent proofs of the Limit Circle Theorem (uniformization
of a closed surface with signature), the Principal Circle Theorem (closed surfaces
with symmetries), and the Retrosection Theorem (Schottky uniformization). Addi-
tionally he showed how to construct a simultaneous uniformization of two surfaces
by means of a quasi-Fuchsian group. Teichmuller theory shows how deformations,
or parameterized families, of discrete groups are related to each other via quasi-
conformal mappings. Implying for instance that the Cantor limit set of a finitely
generated Schottky group is quasiconformally equivalent to the usual middle thirds
Cantor set. As illustrated in Figure 1. these Cantor sets can be quite complicated.
Figure 1. A Cantor limit set of a two-generator Schottky group.
About twenty years ago Sullivan [104] introduced the Beltrami equation into
the theory of holomorphic dynamics in his solution of the Fatou-Julia problem on
Previous Page Next Page