Complete minimal surfaces of finite total curvature on punctured spheres with totally ramified value number greater than $2$

Abstract

Motivated by Osserman's problem on the number $D_g$ of omitted values of the Gauss map of a complete minimal surface with finite total curvature in $\boldsymbol{R}^3$, its totally ramified value number $\nu_g$ (referred to in this paper as the \emph{total weight of totally ramified values}) has attracted significant interest. The value of $\nu_g$ provides more detailed information than the number of omitted values alone. In 2006, Kawakami first found that a minimal surface defined on the three-punctured Riemann sphere, originally constructed by Miyaoka and Sato, satisfies $D_g = 2$ and $\nu_g = 2.5 > 2$. Subsequently, in 2024, Kawakami and Watanabe gave another minimal surface defined on the four-punctured Riemann sphere that also satisfies $D_g = 2$ and $\nu_g = 2.5$. To date, these remain the only two known examples of such surfaces satisfying $\nu_g > 2$. In this paper, we provide a systematic construction of meromorphic functions on punctured Riemann spheres that satisfy $\nu_g > 2$. As a consequence, we obtain the following results for complete minimal surfaces of finite total curvature with $\nu_g = 2.5$ within the topological types of the known examples: (1) For the three-punctured sphere, we prove the uniqueness of Miyaoka--Sato's example. (2) For the four-punctured sphere, we completely determine the surfaces with $D_g=2$ and $\nu_g = 2.5$, which include examples other than Kawakami--Watanabe's one. (3) Furthermore, we construct a new example on the four-punctured sphere satisfying $D_g=1$ and $\nu_g = 2.5$.