Structure and realizability for rational maps

TL;DR

This paper introduces a geometric decomposition of rational maps into 'footballs' to address the Hurwitz existence problem, providing new realizability criteria for branch data and unifying several classical results.

Contribution

It establishes a canonical geometric decomposition of rational maps into 'footballs' and applies this to solve the Hurwitz existence problem for certain branch data.

Findings

Decomposition of pullback metric into finitely many footballs

Proved realizability of branch data under specific conditions

Unified classical results and confirmed a conjecture in a special case

Abstract

We establish a structure theorem for rational maps $f:\overline{\mathbb{C}}\to\overline{\mathbb{C}}$: the pullback metric $f^{*}{\rm d}s_{0}^{2}$ of the standard metric ${\rm d}s_{0}^{2}$ admits a canonical decomposition into finitely many footballs -- Riemann spheres with two antipodal conical singularities of equal angle -- by cutting along a finite set of geodesics. This geometric decomposition provides a new framework for the Hurwitz existence problem. As an application, we prove that a collection $\mathcal{D}$ of $k$ nontrivial partitions of a positive integer $d$ satisfying the Riemann--Hurwitz condition is realizable as the branch datum of a rational map whenever $k>l+1$, where $l$ is the minimum partition length. This unifies the classical results of Thom ($l = 1$), Pakovich ($l = 2$) and Bara\'{n}ski ($k\geq d$), and confirms a conjecture of Zheng in an important special case.