A proof of the Ahlfors surface covering theorem (Q865260)

The author gives a simple topological proof of Ahlfors' theorem for compact Riemann surfaces with boundary; the theorem in question gives a geometric interpretation of Nevanlinna's theory of value distribution of meromorphic maps. The proof given here, while simple and elegant, is not entirely self-contained; it makes use of two analytic facts, the linear isoperimetric equality and a regularity theorem for the boundary of a canonical dissection, presumably taken from [\textit{E. Reyssat}, Quelques aspects des surfaces de Riemann. Progress in Mathematics 77. Birkhäuser (1989; Zbl 0689.30001)].