Quasiconformal mappings

In this subsection all mappings are assumed to orientation-preserving, defined between domains (open, connected sets) of the complex plane. We will use $ w:D\to D'$ for one such mapping. We can define the following differential operators where $ w$ has partial derivatives:

$\displaystyle w_z = \frac{\partial w}{\partial z} =
\frac{1}{2} \left( \frac{\p...
...\left( \frac{\partial w}{\partial x} + i \frac{\partial
w}{\partial y}\right).

We define the complex dilatation of $ w$ at a point $ z_0$, where $ w$ is differentiable with $ Jac(w)(z_0) = \vert w_z(z_0)\vert^2 - \vert w_{\overline{z}}(z_0)\vert^2 \neq 0$, by $ \mu(z_0) = \mu_w(z_0) = w_{\overline{z}}(z_0) / w_z(z_0)$. Observe that $ \vert\mu(z_0)\vert < 1$ (since $ w$ is orientation-preserving) and $ w_z(z_0) \neq 0$.

For a mapping $ w$ and a point $ z$ in $ D$ we can set $ L(z,r) = \max_\zeta
\{\vert w(\zeta) - w(z)\vert; \vert\zeta - z\vert = r\}$ and $ l(z,r) = \min\zeta
\{\vert w(\zeta) - w(z)\vert; \vert\zeta - z\vert = r\}$ (these expressions make sense for small values of $ r$). If $ w$ is $ C^1$ we have that

$\displaystyle \lim_{r\to 0}
\frac{L(z,r)}{l(z,r)} = \frac{1+\vert\mu(z)\vert}{1-\vert\mu(z)\vert}.

This equation motivates our definition of quasiconformal mapping.

Definition 1.19   An orientation-preserving homeomorphism $ w:D\to D'$ between domains of the complex plane is called quasiconformal (qc) if the circular dilatation at $ z$,

$\displaystyle H(z) = \limsup_{r \to 0} \frac{L(z,r)}{l(z,r)}

is bounded on $ D$. If $ H(z) \leq K$ a.e. on $ D$ then we say that $ w$ is $ K$-quasiconformal.

Another equivalent definition is as follows.

Definition 1.20   An orientation-preserving homeomorphism $ w:D\to D'$ between domains of the complex plane is called quasiconformal (qc) if $ w$ has distributional derivatives in $ L^p_{loc}$, for some $ p\geq 1$, (locally in $ L^p$), and $ \vert w_{\overline z}(z)\vert \leq k \vert w_z(z)\vert$ almost everywhere on $ D$, for some $ k$ satisfying $ 0 \leq k < 1$. The map $ w$ is called $ K$-qc for any $ K$ with $ K \geq (1+k)/(1-k)$.

One can easily compute that a holomorphic mapping is $ 1$-qc, or equivalently that its complex dilatation is identically 0 (this is simply the Cauchy-Riemann equations, $ f_{\overline{z}} = 0$).

Prop 1.21   A $ 1$-qc homeomorphism is a biholomorphism.

Prop 1.22   The inverse of a $ K$-qc mapping is also $ K$-qc. The composition of a $ K_1$-qc mapping and a $ K_2$-qc mapping is a $ K_1K_2$-qc mapping.

The existence of qc mappings on the Riemann sphere with a given dilatation (solutions of the Beltrami equation $ w_{\overline{z}} =
\mu w_z$) is an important result extensively used in the theory of Teichmüller spaces.

Theorem 1.23   Given any $ \mu\in L^\infty({\mathbb{C}})$ with $ \vert\vert\mu\vert\vert _\infty < 1$, there exists a unique quasiconformal homeomorphism of $ \widehat{\mathbb{C}}$, with dilatation $ \mu$, and fixing $ \infty$, 0 and $ 1$ (pointwise). We will denote this mapping by $ w^\mu$.

The following stronger result, that the solutions given above depend holomorphically on the dilatation $ \mu$, is known as the measurable Riemann mapping theorem.

Theorem 1.24 (Ahlfors-Bers [3])   For every fixed $ z \in \mathbb{C}$, the mapping $ \mu \mapsto w^\mu(z)$ is holomorphic: if $ \mu$ depends holomorphically on variables $ (t_1,\ldots,t_k) \in {\mathbb{C}}^k$, then $ w^\mu(z)$ is a holomorphic function of the variables $ (t_1,\ldots,t_k)$.

Pablo Ares Gastesi 2005-08-31