[Paper Review] Normal holomorphic curves from parabolic regions to projective spaces

Alexandre Erëmenko|ArXiv.org|Oct 5, 2007

Meromorphic and Entire Functions17 references19 citations

TL;DR

This paper investigates normal holomorphic curves from parabolic regions (e.g., the complex plane or punctured plane) to complex projective spaces, focusing on uniform continuity with respect to the Fubini–Study metric. It establishes a surjectivity result for interpolation of holomorphic curves through prescribed values at discrete sets, using a novel inverse function theorem framework with Lipschitz control and spherical derivative bounds, extending earlier results in Nevanlinna theory and normal families.

ABSTRACT

A holomorphic map from the complex line to a complex projective space is called normal (a. k. a. Brody curve) if it is uniformly continuous from the Euclidean metric to the Fubini--Study metric. The paper contains a survey of known results about such maps, as well as some new theorems.

Motivation & Objective

• To study normal holomorphic curves from parabolic regions (e.g., C or C*) to P^n, defined as uniformly continuous maps from the Euclidean to Fubini–Study metric.
• To address the interpolation problem: given a discrete set E and target values in P^n, can one construct a normal holomorphic curve passing through them?
• To establish a sufficient condition for the existence of such interpolating curves using a modified inverse function theorem with Lipschitz control and uniform bounds on spherical derivatives.
• To extend results from Nevanlinna theory and normal families to the setting of parabolic domains, particularly focusing on the behavior of spherical derivatives and characteristic functions.

Proposed method

• Uses reduced homogeneous representations f = Π ∘ f̃, where f̃: G → C^{n+1} ∖ {0} is holomorphic with no common zeros.
• Employs the spherical derivative f# defined by (f#)^2 = Σ_{i<j} |f_i' f_j - f_i f_j'|^2 / ||f̃||^4 to measure length distortion from Euclidean to Fubini–Study metric.
• Applies the Ahlfors–Shimizu form of the Nevanlinna characteristic: T(r,f) = ∫₀ʳ A(t,f) dt/t, where A(t,f) = (1/π)∫_{|z|≤t} (f#)^2 dz.
• Introduces a new inverse function theorem framework with a right inverse g⁻¹_m defined on balls of radius δ, satisfying Lipschitz condition with constant L.
• Constructs a sequence (aᵏ) in V^E via iterative updates g(aₘᵏ) = P(bₘ, -φₘ(aᵏ⁻¹)), proving convergence via Cauchy estimates and the condition Lδ < ε/(1+ε).
• Uses the key estimate ||φₘ(a′) - φₘ(a″)|| ≤ δ||a′ - a″||_∞ to ensure uniform Lipschitz control of the perturbation term φₘ.

Experimental results

Research questions

• RQ1 Under what conditions does a normal holomorphic curve f: G → P^n exist that interpolates prescribed values at a discrete set E ⊂ G?
• RQ2 Can the spherical derivative f# be uniformly bounded on compact subsets of G to ensure normality of the family of curves?
• RQ3 What is the role of the Fubini–Study metric in characterizing uniform continuity of holomorphic curves from parabolic regions?
• RQ4 How can one construct such interpolating curves using a fixed-point or iterative method with controlled error propagation?
• RQ5 What conditions on the target values and the geometry of the domain ensure surjectivity of the evaluation map on the space of normal curves?

Key findings

• The spherical derivative f# is uniformly bounded on compact subsets of G if and only if the family of holomorphic curves is normal, by the Ascoli–Arzelà theorem.
• For any discrete set E ⊂ C and any assignment b ∈ (P^n)^E, there exists a normal holomorphic curve f: C → P^n such that f|E = b, provided the inverse function framework with Lδ < ε/(1+ε) is satisfied.
• The construction relies on iterative refinement of initial data using right inverses g⁻¹_m with Lipschitz constant L and error control δ, ensuring convergence to a solution.
• The spherical derivative of the constructed curve f is uniformly bounded due to the uniform bound on ||φₘ(a)|| < δ and the Lipschitz property of φₘ.
• The method ensures that f(s, a) = b_s for all s ∈ E, via the identity P(P(b_s, -φ_s(a)), φ_s(a)) = b_s, which holds under the given metric and continuity assumptions.
• The result is robust under small perturbations, as shown by the Cauchy sequence (aᵏ) converging to a solution a ∈ V^E with ||aᵏ - aᵏ⁻¹||_∞ ≤ Lᵏδᵏ.