For a noncompact complex hyperbolic space form of finite volume X=Bn/Γ, we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification 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
Nearby in the stack
X−X
called `canonical radius' of a cusp of
Γ
. 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
Γ
are larger than
r∗
, then there exist symmetric differentials of
X
vanishing at infinity. As a corollary, we show that the cotangent bundle
TX
is ample modulo the infinity if moreover the injectivity radius in the interior of