On Oikawa’s theorem from an algebraic and geometric point of view

In 1956 K. Oikawa proved that a bordered compact Riemann surface X of genus g with k boundary components can be embedded into a closed Riemann surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \widetilde{X} $$ \end{document} of the same genus in such a way that its complement consists in a disjoint union of k discs and every automorphism of X extends to an automorphism of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \widetilde{X} $$ \end{document} . Much later, in 1982, N. Greenleaf and C. L. May mention that the analytical arguments of Oikawa can be extended to the case of nonorientable compact surfaces. Here we give a new algebraic proof, based on the uniformization theorem, of a similar result for Riemann and Klein surfaces, together with a geometric interpretation that relates the geometry of fundamental regions for groups uniformizing the surfaces X and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \widetilde{X} $$ \end{document} .