  For a noncompact complex hyperbolic space form of finite volume
$X=\mathbb{B}^n/\Gamma$, we consider the problem of producing symmetric
differentials vanishing at infinity on the Mumford compactification
$\overline{X}$ of $X$ similar to the case of producing cusp forms on hyperbolic
Riemann surfaces. We introduce a natural geometric measurement which measures
the size of the infinity $\overline{X}-X$ called `canonical radius' of a cusp
of $\Gamma$. The main result in the article is that there is a constant
$r^*=r^*(n)$ depending only on the dimension, so that if the canonical radii of
all cusps of $\Gamma$ are larger than $r^*$, then there exist symmetric
differentials of $\overline{X}$ vanishing at infinity. As a corollary, we show
that the cotangent bundle $T_{\overline{X}}$ is ample modulo the infinity if
moreover the injectivity radius in the interior of $\overline{X}$ is larger
than some constant $d^*=d^*(n)$ which depends only on the dimension.
