Asymptotic behavior of conformal metrics with null Q-curvature

TL;DR

This paper analyzes the asymptotic behavior of conformal metrics with null Q-curvature, revealing bubble formation in higher dimensions as the prescribed curvature varies, extending previous 2D results.

Contribution

It generalizes the understanding of conformal metric behavior with null Q-curvature from 2D to higher dimensions, using higher order inequalities.

Findings

Normalized conformal metrics form a single spherical bubble at low energy levels

The behavior is characterized as the prescribed curvature parameter approaches zero

The results extend previous 2D findings to higher-dimensional cases.

Abstract

We describe the asymptotic behavior of conformal metrics related to the GJMS operator in the null case, as the prescribed Q-curvature $f_0(x) + \lambda$ gradually changes. We show that if one of the maximum points of $f_0$ is flat up to order $n-1$, the normalized conformal metrics in the lowest energy level will form exactly one spherical bubble as $\lambda$ approaches zero using higher order Bol's inequality. This generalizes the result of Struwe (JEMS, 2020) in the two-dimensional case to higher dimensions and helps rule out the slow bubble case discussed by Ng\^o and Zhang (arXiv:1903.12054) to some degree.