Non-compactness of the Prescribed Q-curvature Problem in Large Dimensions

TL;DR

This paper demonstrates that in dimensions 25 and higher, the set of solutions to the prescribed Q-curvature problem on compact manifolds is non-compact, revealing limitations in solution behavior in high dimensions.

Contribution

It proves the non-compactness of the solution set for the prescribed Q-curvature problem in dimensions N ≥ 25, extending understanding of solution structure in high-dimensional conformal geometry.

Findings

Solution set is non-compact for N ≥ 25

High-dimensional behavior differs from lower dimensions

Advances understanding of geometric PDEs in conformal geometry

Abstract

Let $(M, g)$ be a compact Riemannian manifold of dimension $N \geq 5$ and $Q_g$ be its $Q$ curvature. The prescribed $Q$ curvature problem is concerned with finding metric of constant $Q$ curvature in the conformal class of $g$. This amounts to finding a positive solution to \[ P_g (u)= c u^{\frac{N+4}{N-4}}, u>0 {on} M\] where $P_g$ is the Paneitz operator. We show that for dimensions $N \geq 25$, the set of all positive solutions to the prescribed $Q$ curvature problem is {\em non-compact}.