Bloch-Ros principle and its application to surface theory

Abstract

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.