I have the following questiom: let $X$ and $Y$ be two different points (represented by Riemann surfaces) in the Teichmuller space $T_g$ of genus $g \geq 2$ Riemann surfaces. Then of course $X$ and $Y$ are homeomorphic and not bi-holomorphically equivalent. My question is, whether there exists a holomorphic covering from $X$ to $Y.$ Namely, is there a topological covering $p: X \to Y$ which is holomorphic with respec to the complex structures of $X$ and $Y$? Why or why not?

Thanks in advance!