6

2. QUASICONFORMAL MAPPING S

The class of quasiconformal diffeomorphisms is not closed under uniform lim-

its. Thus the generalization to Sobolev spaces is necessary if one is to solve various

extremal problems and make use of limiting processes. After making this gen-

eralization we find the limit of a suitably normalized sequence of quasiconformal

mappings is either quasiconformal or constant. In this setting the class of quasi-

conformal mappings becomes more flexible and has a greater range of applications.

The equivalence between the geometric definition and the analytic definition

was shown by Gehring and Lehto in 1959 [40]. The connection between quasiconfor-

mal mappings, Teichmuller theory and quadratic differentials has been intensively

investigated by Ahlfors-Bers, Reich-Strebel, Lehto and others, see [74] and the

references therein.

2.2. The Beltrami Equation

There is another route to the theory of planar quasiconformal mappings and it

is this route that we will largely focus on in the body of this paper.

Directly from the analytic definition we see that there is a measurable function

ji defined in ft such that

(2.3) df(z) = fi{z) df(z)

where d = \(dx + idy) and d = \(dx — idy). Indeed

(2-4) Moo = | - ^ 1

Equation (2.3) is called the complex Beltrami equation and K provides the elliptic-

ity bounds for this differential equation. Notice that when \i — 0, or equivalently

K — 1, we obtain the usual Cauchy-Riemann system. The Beltrami equation, as

we mentioned earlier, has a long history. Gauss first studied the equation, with

smooth /x, in the 1820's while investigating the problem of existence of isothermal

coordinates on a given surface. The complex Beltrami equation was intensively

studied by Morrey in the late 1930's, and he established the existence of homeo-

morphic solutions for measurable fi [87, 88]. Lehto points out, [74] pp. 24, that

it took another 20 years before Bers recognized that homeomorphic solutions are

quasiconformal mappings in [17].

The function /if = df/df is called the Beltrami coefficient of / or the complex

dilatation of / .

Suppose that / is an orientation preserving linear mapping of C. Then a little

geometric calculation will show that f(z) = az + b~z, \a\ |6|, fif(z) — b/a and /

maps the unit circle to an ellipse and the ratio of the major and minor axes of this

ellipse is

|o| + |6|

=

l + |/y|

\a\-\b\ 1-lM/l

In this way we may view the Beltrami coefficient of a quasiconformal / : ft — C as

defining a measurable ellipse field on the domain ft via the affine approximation to

/ at each point ZQ . Indeed the quasiconformal mapping / is differentiate at almost

every zo e ft and we have

f(z) = z0 + df(z0)(z - z0) + df(z0)(z - z0) + o(\z - z0\)

This point of view is especially common in the field of holomorphic dynamics.