# Concentration phenomena for the fractional $Q$-curvature equation in dimension 3 and fractional Poisson formulas

## TL;DR

This paper investigates the behavior of metrics with prescribed fractional Q-curvature in three dimensions, revealing blow-up phenomena and introducing Poisson-type formulas inspired by conformal geometry and higher-dimensional analysis.

## Contribution

It demonstrates blow-up behavior of fractional Q-curvature metrics in 3D and develops new Poisson-type representation formulas for higher dimensions.

## Key findings

- Metrics can blow up on large sets related to biharmonic functions.
- Constructs explicit examples of blow-up behavior.
- Introduces general Poisson-type formulas for fractional curvature problems.

## Abstract

We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order $3$ in $\R^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of $\R^3$ as the restriction of a metric on $\R^4_+$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $\Phi$ in $\R^4_+$), in analogy with a $4$-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.