The Poisson equation on manifolds with positive essential spectrum

TL;DR

This paper proves the existence of solutions to the Poisson equation on certain Riemannian manifolds with positive essential spectrum, even when Ricci curvature is unbounded below, broadening previous results.

Contribution

It introduces a more general framework for solving the Poisson equation on manifolds with positive essential spectrum, relaxing curvature bounds and spectrum conditions.

Findings

Solutions exist under sharp decay conditions on the source function

Allows Ricci curvature to be unbounded from below

Extends previous results to more general spectral and curvature settings

Abstract

We show existence of solutions to the Poisson equation on Riemannian manifolds with positive essential spectrum, assuming a sharp pointwise decay on the source function. In particular we can allow the Ricci curvature to be unbounded from below. In comparison with previous works, we can deal with a more general setting both on the spectrum and on the curvature bounds.