Blow-up phenomena for the constant Q/R-curvature equation

TL;DR

This paper constructs specific metrics on high-dimensional spheres where solutions to a geometric PDE exhibit blow-up behavior, demonstrating non-compactness in the conformal class with positive scalar and Q/R curvature.

Contribution

It provides explicit examples of metrics and solutions showing blow-up phenomena for the Q/R-curvature equation on spheres of dimension at least 25.

Findings

Constructed a smooth metric on bS^n with non-compact conformal class solutions.

Demonstrated blow-up solutions for the Q/R-curvature equation.

Showed non-compactness of the set of metrics with positive scalar and Q/R curvature.

Abstract

Let $n\ge 25$ be an integer. In this paper, we construct a smooth metric $g_{0}$ on $\mathbb{S}^n$ with the property that the set of metrics in the conformal class of $g_{0}$ having positive scalar curvature and positive constant quotient $Q/R$ is non-compact. Equivalently, we construct families of solutions exhibiting blow-up behavior for the following equation \begin{align*} P _{g_{0}}u- \frac{ (n+2 )(n-4 )}{4} u^{ \frac{2}{n-4}} L_{g_{0}}u^{ \frac{n-2}{n-4}} =0, \quad u>0\quad\text{on} \ \mathbb{S}^{n}, \end{align*} where $P _{g_{0}}$ is the Paneitz operator and $ L_{g_{0}}=-\Delta_{g_{0}} +\frac{n-2}{4(n-1 )}R_{g_{0}} $ is the conformal Laplacian of $ g_{0}$.