Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed

2024/01/17 by Zhang, Yibo
#14 D 06 #30 F 60 #32 G 15 #32 S 50 #57 K 20 #57 K 40 #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2401.09327

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 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 M_h where systole is bounded away from 0, up to equivalence.

Related

• Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space
• Hyperbolic spaces in Teichmüller spaces
• Reduction theory for mapping class groups and applications to moduli spaces
• The kernel of the monodromy of the universal family of degree d smooth plane curves
• The moduli space of Riemann surfaces is Kahler hyperbolic