Papers › Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed
Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed
Yibo Zhang
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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 ℳₕ 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 ℳₕ where systole is bounded away from $0$, up to equivalence.
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