Maps between simply-connected Riemann surfaces Consider the unit disc, complex plane, and Riemann sphere.  For each ordered pair of these surfaces, is there a surjective holomorphic map?  An injective holomorphic map?
 A: *

*Unit disk to complex plane:


*

*Surjective? Consider $\sin((z+1)^{-3})$.  The cubing sends a $120^\circ$ wedge at $-1$ included in the disk onto a punctured neighborhood of $-1$, and $\sin(1/z)$ sends each punctured disk at its essential singularity onto the plane.
Easier would be to use a linear fractional transformation to map the open disk onto the half plane with real part greater than $-1$, then square.

*Injective? Yes.


*Unit disk to Riemann sphere:


*

*Surjective? There is a surjective holomorphic map from the disk to the plane (see above) and from the plane to the sphere (see below), so yes.

*Injective? Yes.


*Complex plane to unit disk:


*

*There are no nonconstant holomorphic functions from the plane to the disk.


*Complex plane to the Riemann sphere:


*

*Surjective? Consider $\frac{z^2}{1-z}$.

*Injective? Yes.


*Riemann sphere to the complex plane or unit disk:


*

*There are no nonconstant holomorphic functions from the Riemann sphere to the complex plane.  Removing $\infty$ from the domain would give an entire function with a removable singularity at $\infty$, which must be constant by Liouville's theorem.  (Or, note that the image of the sphere would be compact, hence bounded, hence the function must be constant by Liouville.)


