This question is obviously broad; turning this broadness into something sharp is part of the problem.

Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what condition are needed to this sequence to converge (uniformely on compact subsets) to a function $h$ holomorphic and proper?

Does exists some statement dealing with this?

This question is obviously broad; turning this broadness into something sharp is part of the problem.

Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what conditions are needed to make this sequence to converge uniformly on compact subsets to a function $h$ holomorphic and proper?

Does there exist some statement/theorem dealing with this?