Structure and realizability for rational maps
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914576919953408 |
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| author | Wei, Zhiqiang |
| author_facet | Wei, Zhiqiang |
| contents | 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 Barański ($k\geq d$), and confirms a conjecture of Zheng in an important special case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06784 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure and realizability for rational maps Wei, Zhiqiang Differential Geometry Algebraic Topology Geometric Topology 57M12 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 Barański ($k\geq d$), and confirms a conjecture of Zheng in an important special case. |
| title | Structure and realizability for rational maps |
| topic | Differential Geometry Algebraic Topology Geometric Topology 57M12 |
| url | https://arxiv.org/abs/2511.06784 |