# Conformal maps between two given domains

Consider two domains \begin{aligned} D_1&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq 0\},\\ D_2&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq \psi(x_1,x_2,...,x_{n-1})\}, \end{aligned} where $$\psi:\mathbb{R}^{n-1}\to\mathbb{R}$$ is a smooth bounded function. I want to construct a conformal map $$\Phi:D_1\to D_2$$. I know that conformal maps between two arbitrary domains in high dimensional Euclidean space may not exist. Nevertheless here the domains are easy, I wonder if I can get such maps. If not, can I put more assumptions on it such that the result is true? Can you give me some hints or references?

• The relevant theorem in this case is Liouville's (see this fairly comprensive Q&A). Commented Jul 24, 2023 at 7:04

Any conformal map in dimensions $$\ge 3$$ is necessary a superposition of inversions and isometries (see e.g. the link suggested by Daniele Tampieri in his comment), so it takes the boundary of $$D_1$$ to a hyperplane or (a part of) a sphere. Therefore, the graph of the $$\psi$$ is a part of a sphere provided a conformal map exists.