Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed

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.