The Poisson equation on complete manifolds with positive spectrum and applications

In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function f has decay f=O(r−1−ε) for some ε>0, where r is the distance function to a fixed point, then the Poisson equation Δu=f has a solution u with at most exponential growth.We apply this result on the Poisson equation to study the existence of harmonic maps between complete manifolds and also existence of Hermitian-Einstein metrics on holomorphic vector bundles over complete manifolds, thus extending some results of Li-Tam and Ni.Assuming moreover that the manifold is simply connected and of Ricci curvature between two negative constants, we can prove that in fact the Poisson equation has a bounded solution and we apply this result to the Ricci flow on complete surfaces.