$L^2$ vanishing theorem on some K\"{a}hler manifolds

Let $E$ be a Hermitian vector bundle over a complete K"{a}hler manifold $(X,\omega )$, ${\dim }_{\mathbb{C}}X=n$, with a $d$(bounded) K"{a}hler form $\omega$, $d_A$ be a Hermitian connection on $E$. The goal of this article is to study the $L^2$-Hodge theory on the vector bundle $E$. We extend the results of Gromov’s \cite{Gro} to the Hermitian vector bundle. At last, as an application, we prove a gap result for Yang-Mills connection on the bundle $E$ over $X$.