Hodge theory of holomorphic vector bundle on compact K\"{a}hler hyperbolic manifold

Let $E$ be a holomorphic vector bundle over a compact K"{a}hler manifold $(X,\omega )$ with negative sectional curvature $sec\le -K<0$, ${\mathrm{\Delta }}_E$ be the Chern connection on $E$. In this article we show that if $C:=\mathrm{\mid }[\mathrm{\Lambda },i\mathrm{\Theta }(E)]\mathrm{\mid }\le c_{n}K$, then $(X,E)$ satisfy a family of Chern number inequalities. The main idea in our proof is study the $L^2$ ${\bar{\mathrm{\partial }}}_{\tilde{E}}$-harmonic forms on lifting bundle $\tilde{E}$ over the universal covering space $\tilde{X}$. We also observe that there is a closely relationship between the eigenvalue of the Laplace-Beltrami operator ${\mathrm{\Delta }}_{{\bar{\mathrm{\partial }}}_{\tilde{E}}}$ and the Euler characteristic of $X$. Precisely, if there is a line bundle $L$ on $X$ such that ${\chi }^p(X,L^{\otimes m})$ is not constant for some integers $p\in [0,n]$, then the Euler characteristic of $X$ satisfies $(-1)^{n}\chi(X)\geq (n+1)+\lfloor\frac{c_{n}K}{2nC} \rfloor$.