arXiv subjects

Shunsuke Kasao

Publications and source records attributed to Shunsuke Kasao.

2 recordsLinked to original sources

arXiv2026

Application of the Bloch--Ros principle to ramification theorem

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.

math.DG↗

arXiv2024

Bloch-Ros principle and its application to surface theory

There exists the duality between normal family theory and value distribution theory of meromorphic functions, which is called the Bloch principle. Zalcman formulated a more precise statement on it. In this paper, based on the Zalcman and Ros work, we comprehend the phenomenon of the trinity among normal family theory, value distribution theory and minimal surface theory and give a systematic description to the relationship among the Montel theorem, the Liuoville theorem and the Bernstein theorem as well as the Carathéodory-Montel theorem, the Picard little theorem and the Fujimoto theorem. We call this phenomenon Bloch-Ros principle. We also generalize the Bloch-Ros principle to various classes of surfaces, for instance, maxfaces in the Lorentz-Minkowski $3$-space, improper affine fronts in the affine $3$-space and flat fronts in the hyperbolic $3$-space. In particular, we give an effective criterion to determine which properties for meromorphic functions that play a role of the Gauss maps of these classes of surfaces satisfy the Gaussian curvature estimate.

math.DG↗