What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$.

The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion.

Specifically, if S contains small sub-squares, they will be mapped into sub-shapes of C that are not squares. The amount of distortion depends on the specific mapping selected, and also on the placement of the small square inside S (see, for example, this nice illustration: the distortion is minimal near the center, and maximal near the corners).

The amount of distortion can be quantified in the following way: Let $s$ be a small sub-square contained in $S$. Let $d(s)$ be its image under the conformal mapping ($d(s)$ is contained in $D$). Let $m(d(s))$ be the maximum-area square that is contained in $d(s)$. Define:

$\mathrm{distortion}(s) = \mathrm{area}(d(s)) / \mathrm{area}(m(d(s))) – 1$

So, if $s$ is mapped to a square, then $d(s)$ is a square, $m(d(s))=d(s)$, and $\mathrm{distortion}(s)=0$.

I would like to know:

  • What is the maximum distortion of a square of a certain size?
  • What is the average distortion, over all squares of a certain size?
  • If we divide $S$ into a k-by-k grid of $k^2$ sub-squares, what is the average distortion over all $k^2$ sub-squares?

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