Structure and realizability for rational maps

We establish a structure theorem for rational maps $f:\bar{\mathbb{C}}\to \bar{\mathbb{C}}$: the pullback metric $f^{\ast }\mathrm{d}{s}_0^2$ of the standard metric $\mathrm{d}{s}_0^2$ 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 Barański ($k\ge d$), and confirms a conjecture of Zheng in an important special case.