"Large" conformal metrics of prescribed Q-curvature in the negative case

TL;DR

This paper proves the existence of at least two distinct conformal metrics with prescribed Q-curvature close to a non-constant function on a four-dimensional manifold with negative total Q-curvature, analyzing their bubbling behavior as a parameter tends to zero.

Contribution

It establishes the existence of multiple conformal metrics with prescribed Q-curvature in the negative case and introduces estimates for large solutions to study bubbling phenomena.

Findings

Existence of at least two conformal metrics for small positive parameters.

Development of estimates for large solutions to analyze bubbling.

Insights into the behavior of solutions as the parameter approaches zero.

Abstract

Given a compact and connected four dimensional smooth Riemannian manifold $(M,g_0)$ with $k_P := \int_M Q_{g_0} dV_{g_0} <0$ and a smooth non-constant function $f_0$ with $\max_{p\in M}f_0(p)=0$, all of whose maximum points are non-degenerate, we assume that the Paneitz operator is nonnegative and with kernel consisting of constants. Then, we are able to prove that for sufficiently small $\lambda>0$ there are at least two distinct conformal metrics $g_\lambda=e^{2u_\lambda}g_0$ and $g^\lambda=e^{2u^\lambda}g_0$ of $Q$-curvature $Q_{g_\lambda}=Q_{g^\lambda}=f_0+\lambda$. Moreover, by means of the "monotonicity trick", we obtain crucial estimates for the "large" solutions $u^\lambda$ which enable us to study their "bubbling behavior" as $\lambda \downarrow 0$.