In the book "Computational Conformal Geometry" by David Gu, the author writes:
"Riemann uniformization theorem states that all surfaces in real life can be conformally mapped to one of three canonical shapes: the unit sphere, the Euclidean plane, and the hyperbolic space."
The the following section he says:
"Surfaces can be classified using conformal geometry. Two surfaces are conformally equivalent if they can be conformally mapped to each other."
Now, what I understand from these sentences would be that, since every surface in 3D can be conformally mapped to either a sphere, a plane or a disc, then these would be the only equivalence classes for conformal equivalence in 3D... This does not seems correct. What am I misunderstanding from these statements?