On Effective Existence of Symmetric Differentials of Complex Hyperbolic Space Forms

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.