Generating all Ahlfors currents by a single entire curve

TL;DR

This work resolves Sibony's conjecture for weak Oka-1 manifolds by constructing an entire curve $F:\mathbb{C}\to X$ whose concentric discs generate all Ahlfors currents on $X$ under the Runge approximation on discs. The authors combine a holomorphic gluing technique, a novel ping-pong induction, and a careful inductive limiting process to amalgamate a countable family of discs into concentric discs from a single $F$, ensuring precise length-area and cohomology control. Central to the approach are a countable disc family generating all Ahlfors currents (via Xie24) and a shrinking-dumbbell lemma with Carathéodory kernel convergence that enables precise current convergence. The result highlights the extraordinary flexibility of entire curves in the weak Oka-1 setting and lays groundwork for potential extensions to Nevanlinna currents and broader Oka-type phenomena.

Abstract

Let \(X\) be a compact complex manifold possessing the \emph{Runge approximation property on discs}, meaning that every holomorphic map from a closed disc into \(X\) is approximable by a global holomorphic map from \(\mathbb{C}\). We construct an entire curve \(F : \mathbb{C} \to X\) such that the associated family of concentric holomorphic discs \(\{F|_{\overline{\mathbb{D}}_r}\}_{r>0}\) generates all Ahlfors currents on \(X\), thereby settling a conjecture of Sibony.

Figures (3)

• Figure 1: Dumbbell-shaped neighborhood of $D_1$ and $D_2$.
• Figure 2: Dumbbell domain convergence.
• Figure 3: Conformal map from dumbbell domain to disc.

Theorems & Definitions (22)

• Conjecture 1.2
• Definition 1.3: Xie24
• Theorem 1.4: Xie24
• Theorem 1.5: WuXie25
• Definition 1.6
• Definition 1.7
• Theorem 1.8: Main Theorem
• Theorem 2.1
• Definition 2.2: cf. FornaessForstnericWold20
• Theorem 2.3: Mergelyan theorem for manifold-valued maps on admissible sets; cf. AlarconForstneric21