Abstract

The "measurable Riemann Mapping Theorem" (or the existence theorem for

quasiconformal mappings) has found a central role in a diverse variety areas such as

holomorphic dynamics, Teichmuller theory, low dimensional topology and geometry,

and the planar theory of PDEs. Anticipating the needs of future researchers we

give an account of the "state of the art" as it pertains to this theorem, that is to

the existence and uniqueness theory of the planar Beltrami equation, and various

properties of the solutions to this equation. The classical theory concerns itself with

the uniformly elliptic case (quasiconformal mappings). Here we develop the theory

in the more general framework of mappings of finite distortion and the associated

degenerate elliptic equations.

We recount aspects of this classical theory for the uninitiated, and then de-

velop the more general theory. Much of this is either new at the time of writing,

or provides a new approach and new insights into the theory. Indeed, it is the

substantial recent advances in non-linear harmonic analysis, Sobolev theory and

geometric function theory that motivated our approach here. The concept of a

principal solution and its fundamental role in understanding the natural domain of

definition of a given Beltrami operator is emphasized in our investigations. We be-

lieve our results shed considerable new light on the theory of planar quasiconformal

mappings and have the potential for wide applications, some of which we discuss.

Received by the editor Received by the editor July 27, 2000, and in revised form September

10, 2004.

2000 Mathematics Subject Classification. Primary 30C60, 35J15, 35J70.

Key words and phrases. Beltrami Equation, quasiconformal, finite distortion, partial differ-

ential equations, degenerate elliptic.

The first author was supported in part by grants from the US National Science Foundation.

The second author was supported in part by the NZ Marsden Fund, the Royal Society of NZ

(James Cook Fellow) and the NSF.