On the totally ramified value number of the Gauss map of minimal surfaces (Q863735)

The Gauss map of a non-flat complete regular minimal surface \(x: M\to\mathbb{R}^4\) in Euclidean space \(\mathbb{R}^3\) is a meromorphic function \(g\) on an open Riemannian surface \(M\). A totally ramified value of \(g\) is either an exceptional value \(a\in\mathbb{C}\cup\{\infty\}\) of \(g\) (being missed by \(g\)) or a value \( \in\mathbb{C}\cup\{\infty\}\) such that \(g\) branches at all the inverse image points of \(b\). Denote by \(D_g\) the number of exceptional values of \(g\). Then the totally ramified value number (TRVN) \(v_g\) of \(g\) is defined by \(v_g= D_g+\Sigma_j(1- 1/n_j)\), where \(n_j\) is the minimum of the multiplicities of \(g\) at the points \(g^{-1}(b_j)\). It is already known that \(D_g\leq v_g\leq 4\) (H. Fujimoto 1992) and, if \(x\) is a complete regular minimal surface with finite total curvature, \(D_g\leq 3\) (R. Ossermann 1964). The author proves that there exist algebraic minimal surfaces having \(v_g= 2.5\) (main theorem) and he shows (to be proved in a forthcoming note) that \(v_g< 4\). An open question is whether an algebraic minimal surface with \(v_g> 2.5\) exists.