  Let $X$ denote a noncompact finite volume hyperbolic Riemann surface of genus
$g\geq 2$, with only one puncture at $i\infty$ (identifying $X$ with its
universal cover $\mathbb{H}$). Let $\overline{X}:=X\cup\lbrace i\infty\rbrace$
denote the Satake compactification of $X$. Let $\Omega_{\overline{X}}$ denote
the cotangent bundle on $\overline{X}$. For $k\gg1$, we derive an estimate for
$\mu_{\overline{X}}^{\mathrm{Ber},k}$, the Bergman metric associated to the
line bundle $\mathcal{L}^{k}:=\Omega_{\overline{X}}\otimes
\mathcal{O}_{\overline{X}}\big((k-1)\infty\big)$.
  For a given $d\geq 1$, the pull-back of the Fubini-Study metric on the
Grassmannian, which we denote by
$\mu_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$, defines a K\"ahler metric
on $\mathrm{Sym}^d(\overline{X})$, the $d$-fold symmetric product of
$\overline{X}$. Using our estimates of $\mu_{\overline{X}}^{\mathrm{Ber},k}$,
as an application, we derive an estimate for
$\mu_{\mathrm{Sym}^d(\overline{X}),\mathrm{vol}}^{\mathrm{FS},k}$, the volume
form associated to the (1,1)-form
$\mu_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$.
