Does a conformal map between two bounded convex surfaces exist generally?

Question

Suppose we have two bounded surfaces, which are both flat or convex, is there a conformal map that links the two surfaces in general, such as $x^2+y^2<a^2, z=0$ and $z=\sqrt{a^2-x^2-y^2}$?

Answer 1

The two surfaces should be the same topological type, and if that topological type is disk or sphere than the answer is yes, otherwise, not so much.

Answer 2

The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to either the unit disk, the whole plane, or the sphere. The question whether it is the sphere is easy to decide by topology, the question whether it is the disk or the plane is harder. E.g., a punctured sphere is conformally equivalent to the plane, a hemisphere is conformally a disk. (In these two examples it is easy to check that the uniformizing map is stereographic projection.)