Recently I met the following result:
A holomorphic map from $\mathbb{C}$ to a compact Riemann surface $M$ is a constant map.
The proof is given as follows:
Suppose a holomorphic map $f:\mathbb{C}\to M$, since $\mathbb{C}$ is simply connected and $\pi: \mathbb{H}\to M$ is a universal covering, there is a lift $\tilde{f}:\mathbb{C}\to \mathbb{H}.$ By Liouville Theorem, $\tilde{f}$ is constant. So is $f$.
Why is $\tilde{f}$ bounded? I am confused about this. Any help will be appreciated.
You cannot prove directly that $\tilde{f}$ is bounded. But it is a constant because $e^{i\tilde{f}}$ is bounded. Aplly Liouville's Thereom to this function.