Bloch-Ros原理及其在曲面理论中的应用
Bloch-Ros principle and its application to surface theory
AI总结：
本文基于Zalcman和Ros的工作提出Bloch-Ros原理，揭示了正规族理论、值分布理论与极小曲面理论间的三位一体现象，并将其推广至多种曲面类，给出了判定亚纯函数高斯曲率估计的有效准则。
摘要：
亚纯函数的正规族理论与值分布理论之间存在对偶性，这被称为Bloch原理。Zalcman对此给出了更精确的表述。在本文中，基于Zalcman和Ros的工作，我们理解了正规族理论、值分布理论与极小曲面理论之间的三位一体现象，并系统描述了Montel定理、Liuoville定理与Bernstein定理，以及Carathéodory-Montel定理、Picard小定理与Fujimoto定理之间的关系。我们将这种现象称为Bloch-Ros原理。我们还将Bloch-Ros原理推广到各类曲面，例如Lorentz-Minkowski $3$空间中的maxfaces、仿射$3$空间中的不当仿射波前以及双曲$3$空间中的平坦波前。特别地，我们给出了一个有效准则，用于确定作为这些曲面类高斯映射的亚纯函数的哪些性质满足高斯曲率估计。
英文摘要：
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.
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