{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/holomorphic-curves-in-moduli-spaces-are-quasi","title":"Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed","arxiv_id":"2401.09327","date":"2024-01-17","proceeding":null,"authors":["Yibo Zhang"],"abstract":"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 Teichm\\\"uller 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.","url_abs":"https://arxiv.org/abs/2401.09327v1","url_pdf":"https://arxiv.org/pdf/2401.09327v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"holomorphic-curves-in-moduli-spaces-are-quasi","repo_url":"https://github.com/ahdoc/hurwitzmoves_to_algintersections","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}