The Poisson equation on complete manifolds with positive spectrum and applications

TL;DR

This paper studies solutions to the Poisson equation on complete manifolds with positive spectrum, establishing existence results under decay conditions and applying these to harmonic maps, Hermitian Einstein metrics, and Ricci flow.

Contribution

It extends previous results by proving existence of solutions with controlled growth on manifolds with positive spectrum and Ricci curvature bounds, and applies these to geometric analysis problems.

Findings

Solutions exist with at most exponential growth under decay conditions.

Bounded solutions exist on simply connected manifolds with Ricci curvature bounds.

Applications include harmonic maps, Hermitian Einstein metrics, and Ricci flow analysis.

Abstract

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-\varepsilon}) $ for some $\varepsilon >0,$ where $r$ is the distance function to a fixed point, then the Poisson equation $\Delta u=f$ 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 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.