How can I prove the existence or not of a surjective morphisms of Riemann surfaces $\Bbb P^1 \to \Bbb C$ ? and $\Bbb C \to \Bbb P^1$? In case that it exists, how can I prove if there's any that can be extended to a morphism $\Bbb P^1\to \Bbb P^1$? $\Bbb P^1$ is the complex projective space. Morphism here means a holomorphic maps between Riemann surfaces
It should be enough to find and example to prove the existance but I can't come up with any.