Application of the Bloch--Ros principle to ramification theorem

Abstract

The author and Kawakami revealed that the Picard little theorem, the Carathéodory--Montel theorem and the Fujimoto theorem -- phenomena concerning omitted values in value distribution theory, normal family theory and the theory of Gauss maps of minimal surfaces, respectively -- are not isolated results but can be discussed within a unified framework. We call this theoretical framework the Bloch--Ros principle. Furthermore, the value-distribution-theoretic properties of Gauss maps hold not only for minimal surfaces but also for other classes of surfaces that admit singularities, such as maxfaces in the Lorentz--Minkowski $3$-space and improper affine fronts in the affine $3$-space, thereby further extending this theoretical framework. In this paper, we provide a unified approach to phenomena concerning totally ramified values among value distribution theory, normal family theory and the theory of Gauss maps of these classes of surfaces.