Toward Nevanlinna-galois Theory for Algebraic Minimal Surfaces

A complete minimal surface in R is said to be pseudo-algebraic, if its Weierstrass data are defined on a compact Riemann surface finitely many points removed M = M\{P1, . . . , Pn} and extend meromorphically across punctures ([KKM], see also [G]). The punctured Riemann surface M on which the Weierstrass data are defined is called the basic domain. The difference between a complete minimal surface being algebraic or pseudo-algebraic lies in whether the period condition [P] is required or not. It is well known that the period condition is always a very hard obstacle against the attempt constructing algebraic minimal surfaces. We write the Weierstrass data of the minimal surface under consideration as (g, ω). Then g : M → P is the Gauss map. It is then a fundamental question to ask what is P\g(M).