# Families of proper holomorphic maps

## TL;DR

This paper constructs continuous families of proper holomorphic maps from surfaces with varying complex structures to the complex plane.

## Contribution

A novel construction of continuous families of proper holomorphic maps depending on varying complex structures.

## Key findings

- A continuous family of proper holomorphic maps is constructed for surfaces with parameterized complex structures.
- The maps depend continuously on the parameter in a metrisable space.

## Abstract

Given a smooth, open, oriented surface X endowed with a family of complex structures \documentclass[12pt]{minimal}
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\begin{document}$$\{J_b\}_{b\in B}$$\end{document}{Jb}b∈B depending continuously on the parameter b in a metrisable space B, we construct a continuous family of proper holomorphic maps \documentclass[12pt]{minimal}
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\begin{document}$$F_{b}:(X,J_b)\rightarrow \mathbb {C}^{2}$$\end{document}Fb:(X,Jb)→C2, \documentclass[12pt]{minimal}
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\begin{document}$$b\in B$$\end{document}b∈B.