Can Liouville's theorem be proven as a elementary corollary of the fact that regular functions on projective space are constant?

Question

These are so tantalizingly similar that I can't help but wonder if there's a proof of Liouville's theorem that develops enough analysis to state that holomorphic functions are analytic, then switches over to relatively low level (in particular, no GAGA) algebraic geometry to finish things off.

Answer 1

$\def\CC{\mathbb{C}}\def\PP{\mathbb{P}}$Does this satisfy? Let $f : \CC \to \CC$ be a bounded entire function. By Riemann's removable singularity theorem, $f$ extends to a holomorphic function $\CC \PP^1 \longrightarrow \CC$. Since $\CC \PP^1$ is compact, $|f|$ must have a local maximum on $\CC \PP^1$. But, since $f$ is holomorphic, the maximum modulus principle shows that $f$ must be locally constant near that local maximum and, since holomorphic functions are analytic, that means $f$ is constant.

Of the three linked results above, I would judge the removable singularity theorem to be the hardest. Liouville's theorem is easy enough that, by time you have the toolkit to prove the removable singularity theorem, it is no effort to write down the standard proof of Liouville, but I don't think there is any circularity here.