L2 vanishing theorem on some Kähler manifolds

Let E be a Hermitian vector bundle over a complete Kähler manifold (X, ω), dimℂX = n, with a d(bounded) Kähler form ω, and let dA be a Hermitian connection on E. The goal of this article is to study the L2-Hodge theory on the vector bundle E. We extend the results of Gromov [18] to the Hermitian vector bundle. Finally, as an application, we prove a gap result for the Yang-Mills connection on the bundle E over X.