Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Let $X$ denote a noncompact finite volume hyperbolic Riemann surface of genus $g\ge 2$, with only one puncture at $i\mathrm{\infty }$ (identifying $X$ with its universal cover $\mathbb{H}$). Let $\bar{X}:=X\cup \{i\mathrm{\infty }\}$ denote the Satake compactification of $X$. Let ${\mathrm{\Omega }}_{\bar{X}}$ denote the cotangent bundle on $\bar{X}$. For $k\gg 1$, we derive an estimate for ${\mu }_{\bar{X}}^{\mathrm{B}\mathrm{e}\mathrm{r},k}$, the Bergman metric associated to the line bundle ${\mathcal{L}}^k:={\mathrm{\Omega }}_{\bar{X}}\otimes {\mathcal{O}}_{\bar{X}}((k-1)\mathrm{\infty })$. For a given $d\ge 1$, the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by ${\mu }_{{\mathrm{S}\mathrm{y}\mathrm{m}}^d(\bar{X})}^{\mathrm{F}\mathrm{S},k}$, defines a Kähler metric on ${\mathrm{S}\mathrm{y}\mathrm{m}}^d(\bar{X})$, the $d$-fold symmetric product of $\bar{X}$. Using our estimates of ${\mu }_{\bar{X}}^{\mathrm{B}\mathrm{e}\mathrm{r},k}$, as an application, we derive an estimate for ${\mu }_{{\mathrm{S}\mathrm{y}\mathrm{m}}^d(\bar{X}),\mathrm{v}\mathrm{o}\mathrm{l}}^{\mathrm{F}\mathrm{S},k}$, the volume form associated to the (1,1)-form ${\mu }_{{\mathrm{S}\mathrm{y}\mathrm{m}}^d(\bar{X})}^{\mathrm{F}\mathrm{S},k}$.