The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions.


I’m interested in how it generalizes for fractional dimensional sets particular with dimensions between 2 and 3.


We can start by considering dimensions between 1 and 2. If you take a simply connected fractal such as the vicsek fractal with dimension $\log\_3{5}$ then it’s easy to see (though perhaps hard to prove) that any simply connected open subset of this can be invertibly continuously mapped to any other simply connected open subset. Moreover any such map trivially must be conformal since the only angles of intersection possible are right angles and a continuous map would send intersections to intersections.


Based on the idea that conformal maps exist between simply connected open subsets for the prototypical spaces of dimension 1, $\log\_3(5)$ and 2 I’m willing to wager \$10 that such maps exist for all dimensions between 1 and 2. I’d be shocked if a simply connected set with Hausdorff dimension < 2 failed to have a Riemann mapping theorem analogue.


Where it starts to get interesting for me is dimensions greater than 2 but less than 3.


Does anyone know of any results in this space for sets with dimensions between 2 and 3



Do we have uniformization theorems for fractional dimensional spaces?
So I believe the situation is a bit complex. I think Vicsek-like fractals made from squares, cubes, hypercubes etc... all support a riemann mapping theorem and respectively have dimensions $\log\_3(5), \log\_3(7), \log\_3(9) ... \log\_3(2n+1) ... $ which grows without bound.


It seems like dimension isn't really the important thing here. It's rather the "shape of the set" which decides whether conformal mapping between open sets is possible or not. Its weird to write it but "euclidean space is quite pathological compared to vicsek like spaces and possibly other fractals killing off Riemann mapping theorems"


