Is there a conformal mapping (with certain singularities noted below) from the surface of a cube to the surface of a spherical cube that preserves edges? Note that this also implies that vertices and faces will be preserved, of course.
The necessary singularities are the vertices, due to the different angle defects. I am looking for a mapping where the vertices are the only singularities, and the rest of the map is conformal.
It seems like such a mapping should be possible. By the uniformization theorem, we know that a conformal map from a cube with rounded edges to a sphere exists. Additionally, we know that a conformal map from a flat 4-gonal dihedron to a spherical 4-gonal dihedron exists, using the Peirce quincuncial projection (with singularities at the vertices).
I found some sources online claiming that such a map exists, but with no explicit formulas, just some images:
A global shallow‐water model using an expanded spherical cube: Gnomonic versus conformal coordinates (1996) may help you.
I implemented the mapping based on the paper: Mercator projection of the mapping