Bloch–Ros Principle and its Application to Surface Theory

This paper establishes a profound unifying framework, termed the Bloch-Ros Principle, which connects seemingly disparate areas of complex analysis and differential geometry. Traditionally, the Bloch Principle highlights a duality between normal family theory and value distribution theory for meromorphic functions. This work generalizes this concept to encompass surface theory, specifically relating properties of the Gauss map of a surface to its geometric characteristics, such as curvature estimates and global properties.

The significance of this paper lies in its ability to systematize and unify a wide range of classical results across these fields. It meticulously describes relationships among key theorems such as the Montel Theorem, Liouville Theorem, Bernstein Theorem, Carathéodory-Montel Theorem, Picard’s Little Theorem, and Fujimoto’s Theorem, placing them all under a single, coherent theoretical umbrella. This unification provides a powerful general method for deriving geometric properties of surfaces from complex-analytic properties of their Gauss maps.

The key innovations introduced are:

1. Formalization of the Bloch-Ros Principle: The paper generalizes the original Bloch principle, which was initially refined by Zalcman using his effective normality lemma. The authors extend Ros’s application of this idea to minimal surfaces to a much broader class of surfaces.
2. Introduction of the Weierstrass m-triple: This new concept (Definition 3.1) provides a generalized data structure (Σ, f dz, g) for various types of surfaces, where Σ is a Riemann surface, f dz is a holomorphic 1-form, g is a meromorphic function (playing the role of the Gauss map), and m is a positive integer. This allows for a unified treatment of minimal surfaces, maxfaces, improper affine fronts, and flat fronts.
3. Definition of m-curvature estimate: A notion of curvature estimate (Definition 3.3) is introduced, tailored to the Weierstrass m-triples, which quantifies how the Gaussian curvature of the surface behaves under certain conditions.
4. The Central Criterion (Theorem 3.4): This is the paper’s most significant innovation. It provides an effective criterion stating that for a “compact property” P (a topological property of families of holomorphic maps, building on Zalcman’s work) of the generalized Gauss map g, either the m-curvature estimate is satisfied for any Weierstrass m-triple whose g has property P, or there exists a specific “counterexample” surface (a complete disk with a non-constant g having property P and specific bounded curvature). This theorem offers a powerful tool to prove Liouville-type and Picard-type theorems for these generalized surface classes.

The main prior ingredients needed for this work are drawn from both complex analysis and differential geometry:

• From Complex Analysis:
  • Normal Family Theory: The foundational concept of normal families of meromorphic functions (Marty’s theorem) and its associated theorems like Montel’s theorem.
  • Value Distribution Theory: Picard’s Little and Great Theorems and Liouville’s Theorem, which deal with the values omitted by meromorphic functions.
  • Bloch Principle and Zalcman’s Lemma: The original Bloch Principle stating a connection between function properties and normality, and Zalcman’s effective criterion (a key lemma for determining normality), which is central to the “compact property” definition in this paper.
• From Differential Geometry:
  • Minimal Surface Theory: The classical Weierstrass representation for minimal surfaces, the Bernstein theorem (any entire minimal graph in R³ is a plane), and the Fujimoto theorem (complete minimal surfaces whose Gauss maps omit sufficiently many points are planes). These results serve as the primary motivation and classical examples for the generalized Bloch-Ros principle.
  • Conformal Metrics and Gaussian Curvature: The underlying geometric concepts of Riemann surfaces, conformal metrics, and Gaussian curvature, which are essential for defining and analyzing the m-curvature estimate.
  • Specific Surface Classes and their Representations: The generalized Weierstrass-type representations for various surfaces like maxfaces in Lorentz-Minkowski space, improper affine fronts in affine space, and flat fronts in hyperbolic space.

By integrating these advanced mathematical concepts, the paper offers a unified and robust methodology to explore the interplay between complex function theory and surface geometry, extending established dualities to a broader and more diverse landscape of geometric objects.