Structure and realizability for rational maps

TL;DR

This work addresses the realizability of candidate branch data for rational maps on the sphere by marrying Hurwitz theory with differential-geometric methods. A central structure theorem shows that the pullback of the spherical metric under a rational map decomposes into a canonical assembly of footballs (constant curvature $1$ metrics with conical singularities), enabling a constructive gluing approach to realize prescribed branch data. The authors prove realizability under a broad condition $k\ge|\pi_k|+2$ (and relate it to Zheng's conjecture), providing explicit geometric constructions and several examples. The results bridge combinatorial Hurwitz data with geometry, offering a practical, geometric pathway to the Hurwitz existence problem on $S^2$ and extending to broader meromorphic settings.

Abstract

We prove that if $\mathcal{D}$ is a collection of $k$ nontrivial partitions of a positive integer $d$ satisfying the Riemann-Hurwitz formula, and if $k$ is greater than one plus the minimum length among the partitions in $\mathcal{D}$, then $\mathcal{D}$ is realizable as the branch data of a rational map $f \colon S^2 \to S^2$. As an application, we confirm a conjecture of Zheng [\textit{Topology Appl.} \textbf{153} (2006), no.~12, 2124--2134].

Figures (13)

• Figure 1: Constructing an American $S^{2}_{\{\alpha,\alpha\}}$ from the standard football.
• Figure 2: Integral curves
• Figure 3: Construction of $f(z) = \frac{(z - a)^2}{(z - b_1)(z - b_2)}$.
• Figure 4: Construction of $f(z) = \frac{(z - a_1)(z - a_2)}{(z - b_1)(z - b_2)}$.
• Figure 5: Construction of $f(z) = \frac{(z - a)^2}{(z - b)^2}$.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.1: Prime degree conjecture
• Conjecture 1.2
• Theorem 1.3: Structure theorem for rational maps
• Remark 1
• Theorem 1.4
• Remark 2
• Theorem 1.5
• proof