Families of proper holomorphic maps

TL;DR

This work proves the existence of a continuous family of fiberwise proper holomorphic maps from a varying family of Riemann surfaces $(X,J_b)$ to $\mathbb{C}^2$, parameterized by a metrisable space $B$. The authors construct a map $F: B\times X\to C^2$ whose fibers $F(b,\cdot)$ are $J_b$-holomorphic and proper, by a convergent Runge-approximation scheme on an exhaustion by Runge compacta and a parameterized Mergelyan theorem for proper families. They further show how to obtain a continuous family of proper holomorphic immersions into $\mathbb{C}^3$ by augmenting the target with an auxiliary holomorphic component, and they derive a proper harmonic-map analogue. The results extend the program initiated by Forstneric (2025) to deform complex structures while maintaining global holomorphic geometry, with potential implications for analytic subvarieties and Oka-type phenomena.

Abstract

Given a smooth, open, oriented surface $X$ endowed with a family of complex structures $\{J_b\}_{b\in B}$ depending continuously on the parameter $b$ in a metrisable space $B$, we construct a continuous family of proper holomorphic maps $F_{b}:(X,J_b)\to\mathbb C^{2}$, $b\in B$.

Figures (4)

• Figure 3.1: Pictures of sets from Definition \ref{['Definition Pictures of sets']}
• Figure 3.2: Regions in the inductive step in the case $n=2$
• Figure 3.3: Critical case 1
• Figure 3.4: Critical case 2

Theorems & Definitions (28)

• Theorem 1.1
• Corollary 1.2
• proof
• Theorem 1.3
• Corollary 1.4
• proof
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4