There is a conformal map between any two simply connected open proper subsets of the complex plane. This means, for example, there is a one-to-one analytic map from the interior of a square onto the interior of a a circle. Or from the interior of a triangle onto the interior of a pentagon. Or from the Mickey Mouse logo to the Batman logo (see here).
So we can map (the interior of) a rectangle conformally onto a very different shape. Can we map a rectangle onto a rectangle? Yes, clearly we can do this with a linear polynomial, f(z) = az + b. Are there any other possibilities? Surprisingly, the answer is no: if an analytic function takes any rectangle to another rectangle, that analytic function must be a linear polynomial.
Since a linear polynomial is the composition of a scaling, a rotation, and a translation, this says that if a conformal map takes a rectangle to a rectangle, it must take it to a similar rectangle.
These statements are proved in [1]. Furthermore, the authors prove that “An analytic function mapping some closed convex n-gon R onto another closed convex n-gon S is a linear polynomial.”
More posts on conformal mapping
[1] Joseph Bak and Pisheng Ding. Shape Distortion by Analytic Functions. The American Mathematical Monthly. Feb. 2009, Vol. 116, No. 2.
There are some conformal mappings from the circle to itself (the symmetries of the hyperbolic plane). By conjugating these with the mapping between the rectangle and the circle, shouldn’t we get some non-trivial conformal mappings sending the rectangle to itself?
I feel like I’m missing something. If I read this right:
1) Any two rectangles have a conformal map between them.
2) Any conformal map between rectangles is affine linear.
3) Affine linear maps take rectangles to similar rectangles.
So in my mind 1 + 2 + 3 => all rectangles are similar, which is clearly false. There must be a nuance in one of the steps I’m not seeing.
I think the source of confusion is interior versus boundary.
You can map the interior of any rectangle conformally to the interior of another, but this map cannot extend to include the boundaries unless the rectangles are similar. So it’s not the case that any two rectangles have a conformal map between them, if you include the boundary of the rectangle.
And even if you’re mapping the interior of one rectangle to the interior of the other, it’s also the case that no rectangle in the interior goes to a rectangle inside the other, unless the conformal map is a linear polynomial.