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 Δ u = 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. © 2009 Elsevier Inc. All rights reserved.
Munteanu, O., & Sesum, N. (2010). The Poisson equation on complete manifolds with positive spectrum and applications. Advances in Mathematics, 223(1), 198–219. https://doi.org/10.1016/j.aim.2009.08.003