# 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.

## Key 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.