nLab quasiconformal map

Definition

Consider

• open subsets$\;D, D' \subset \mathbb{C}$ of the complex plane,

• a differentiable homeomorphism $f \colon D \longrightarrow D'$.

The ratio

(1)$$\mu_f \coloneqq \frac { \bar\partial f } { \partial f }$$

of the function’s (anti-)holomorphic derivatives

$$\partial f \,\coloneqq\, \frac{\partial f}{\partial z} \;\coloneqq\; \tfrac{1}{2}\big( \tfrac{\partial f}{\partial x} -\mathrm{i} \tfrac{\partial f}{\partial y} \big) \,,\;\;\; \bar\partial f \,\coloneqq\, \frac{\partial f}{\partial \bar z}$$

is called the complex dilatation $\mu_f$ of $f$, a measure for the function’s failure to be a conformal map or holomorphic map.

The actual dilatation of $f$ is defined to be the ratio

(2)$$D_f \;\coloneqq\; \frac { 1 + d_f } { 1 - d_f }$$

for

$$d_f \;\coloneqq\; \left\vert\mu_f\right\vert = \frac { \left\vert \bar\partial f\right\vert } { \left\vert \partial f\right\vert }$$

the absolute value of the complex dilation (1).

The function $f$ is called quasi-conformal if its dilatation $D_f \colon \mathbb{C} \longrightarrow \mathbb{R}$ (2) is a bounded function.

References

• Lars V. Ahlfors; pp. 4 of: Lectures on quasiconformal mappings, Van Nostrand, Princeton (1966), University Lecture Series 38 AMS (2006) [ams:ULECT/38, pdf]

• Davoud Cheraghi; sections 7.1-2 in: Geometric Complex Analysis (2016) [pdf, Cheraghi-ComplexAnalysis.pdf?]

• Nikolai V. Ivanov: The geometric meaning of the complex dilatation [arXiv:1701.06259]

See also:

• Wikipedia: Quasiconformal map