Because the Riemann mapping theorem does not hold in higher dimensions. While there are all sorts of conformal mappings in dimension 2, for higher dimensions <a href="https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)">Liouville's theorem</a> restricts all possible conformal mappings to the ones that are compositions of similarities, translations, orthogonal transformations and inversions. In generality there are contractible spaces not homeomorphic (therefore not conformal) to $\mathbb{R}^n$ such as <a href="https://en.wikipedia.org/wiki/Whitehead_continuum">Whitehead continuum</a>

As for the proof of Liouville's theorem, maybe <a href="https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-28/issue-2/Liouvilles-theorem/pjm/1102983461.full"> this article</a> is of interest where you can see a sketch of Nevanlinna's original proof and a proof by nonstandard analysis.