# On the prescribed $Q$-curvature problem in Riemannian manifolds

## TL;DR

This paper establishes conditions under which metrics with prescribed Q-curvature exist on Riemannian manifolds, including special cases in four dimensions and for open submanifolds, broadening the understanding of curvature prescription problems.

## Contribution

It provides new existence results for prescribed Q-curvature under natural assumptions, including in four dimensions and for open submanifolds, with minimal restrictions on the functions involved.

## Key findings

- Existence of metrics with prescribed Q-curvature under natural sign conditions.
- Results for four-dimensional manifolds with restrictions on Euler characteristic.
- Any smooth function on r^n can be realized as Q-curvature of some metric.

## Abstract

We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on $\mathbb{R}^n$ can be realized as the $Q$-curvature of a Riemannian metric.