Estimates and asymptotics of Teichmüller modular forms

Abstract: In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}{g}$ be the Deligne-Mumford compactification of $\mathcal{M}{g}$, and we denote its boundary by $\partial\mathcal{M}{g}$. Let $\pi:\mathcal{C}{g}\longrightarrow\mathcal{M}{g}$ be the universal surface. For any $n\geq 1$, let $\Lambda_{n}:=\pi_{\ast}(T_{v}\mathcal{C}{g})^{n}$, where $T{v}\mathcal{C}{g}$ denotes the vertical holomorphic tangent bundle of the fibration $\pi$, and the fiber of $\Lambda{n}$ over any $X\in\mathcal{M}{g}$ is equal to $H^{0}(X,\Omega{X}^{\otimes n})$, the space of holomorphic differentials of degree-$n$, defined over the Riemann surface $X$. Let $\lambda_{n}:=\mathrm{det}(\Lambda_{n})$ denote the determinant line bundle of the vector bundle $\Lambda_{n}$, whose sections are known as Teichm\"uller modular forms. The complex vector space of Teichm\"uller modular forms is equipped with Quillen metric, which is denoted by $|\cdot|_{\mathrm{Qu}}$.