I stumbled upon the following remark somewhere and unfortunately lacks a proof. "Every Riemann Surface in $\mathbb{C}^{2}$ in non-compact". A complex algebraic curve given as a zero locus of polynomial $P \in \mathbb{C}[x,y]$, apparently is not compact (since it's not bounded). But what about other Riemann Surfaces in $\mathbb{C}^{2}$, can you give me a reference or a proof of the aforementioned statement? Moreover, does this sentence mean that every Riemannian Surface cannot admit a holomorphic embedding into $\mathbb{C}^{2}$ even if it is compact on its own as a topological space?
P.S. If I have said something completely wrong or mistaken please do let me know. I'm not very familiar with all these things. Thank you!