Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916946763579392 |
|---|---|
| author | Zhang, Yibo |
| author_facet | Zhang, Yibo |
| contents | A holomorphic curve in moduli spaces is the image of a non-constant holomorphic map from a hyperbolic surface $B$ of type $(g,n)$ to the moduli space $\mathcal{M}_h$ of closed Riemann surfaces of genus $h$. We show that, when all peripheral monodromies are of infinite order, the holomorphic map is a quasi-isometric immersion with parameters depending only on $g$, $n$, $h$ and the systole of $B$. When peripheral monodromies also satisfy an additional condition, we find a lift quasi-isometrically embedding a fundamental polygon of the hyperbolic surface $B$ into the Teichmüller space. We further improve the Parshin-Arakelov finiteness theorem, by proving that there are only finitely many monodromy homomorphisms induced by holomorphic curves of type $(g,n)$ in $\mathcal{M}_h$ where systole is bounded away from $0$, up to equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09327 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed Zhang, Yibo Geometric Topology Metric Geometry 14 D 06, 30 F 60, 32 G 15, 32 S 50, 57 K 20, 57 K 40 A holomorphic curve in moduli spaces is the image of a non-constant holomorphic map from a hyperbolic surface $B$ of type $(g,n)$ to the moduli space $\mathcal{M}_h$ of closed Riemann surfaces of genus $h$. We show that, when all peripheral monodromies are of infinite order, the holomorphic map is a quasi-isometric immersion with parameters depending only on $g$, $n$, $h$ and the systole of $B$. When peripheral monodromies also satisfy an additional condition, we find a lift quasi-isometrically embedding a fundamental polygon of the hyperbolic surface $B$ into the Teichmüller space. We further improve the Parshin-Arakelov finiteness theorem, by proving that there are only finitely many monodromy homomorphisms induced by holomorphic curves of type $(g,n)$ in $\mathcal{M}_h$ where systole is bounded away from $0$, up to equivalence. |
| title | Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed |
| topic | Geometric Topology Metric Geometry 14 D 06, 30 F 60, 32 G 15, 32 S 50, 57 K 20, 57 K 40 |
| url | https://arxiv.org/abs/2401.09327 |