Consider two domains $$ \begin{aligned} D_1&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq 0\},\\ D_2&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq \psi(x_1,x_2,...,x_{n1})\}, \end{aligned} $$ where $ \psi:\mathbb{R}^{n1}\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?
1 Answer
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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.

1$\begingroup$ Good point. (But a sphere isn't possible, so the graph of đťś“ must be a hyperplane.) $\endgroup$ Commented Jul 24, 2023 at 15:45