I have been reading Choi-Schoen's 'The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature' and in the proof of proposition 1, they mentioned the following fact which I have difficulty with understanding. Let M be an orientable, minimal (which I don't think that we need for the questions I will ask) surface with smooth compact boundary and finite Euler characteristic then M is conformally a closed Riemann Surface with a finite number of disks and points are removed.
In Meeks and Perez's monograph[p18] they explained a similar result for complete minimal surfaces by using the fact that the Gauss map is conformal.
I am confused with handling the boundary components of M. Any help would be appreciated.