Normal conformal metrics with prescribed $Q$-Curvature in $\mathbb{R}^{2n}$

TL;DR

The paper studies the prescribed $Q$-curvature equation $(-Δ)^n u = K(x) e^{2n u}$ in ${\mathbb R}^{2n}$, linking normal conformal metrics with finite total curvature to the data $K$. It develops a linear theory for the polyharmonic operator, derives precise asymptotics at infinity via a representation and a Pohozaev identity, and obtains a necessary condition on the total curvature $\Lambda_u$ for normal metrics when $K$ has mild growth. Existence results are proved for nonpositive $K$ with polynomial growth, and in the radial setting without growth assumptions; these results use Leray-Schauder fixed point theory and detailed radial linear theory to construct and control solutions. The work extends higher-dimensional conformal geometry by clarifying when normal metrics exist and how their total curvature interacts with the prescribed $Q$-curvature and radial symmetry.

Abstract

We consider the $Q$-curvature equation \begin{equation}\label{0.1} (-Δ)^n u = K(x)e^{2nu}\quad\text{in} ~\mathbb{R}^{2n} \ (n \geq 2) \end{equation} where $K$ is a given non constant continuous function. Under mild growth control on $K$, we get a necessary condition on the total curvature $Λ_u$ for any normal conformal metric $g_u = e^{2u}|dx|^2$ satisfying $Q_{g_u} = K$ in $\mathbb{R}^{2n}$, or equivalently, solutions to equation with logarithmic growth at infinity. Inversely, when $K$ is nonpositive satisfying polynomial growth control, we show the existence of normal conformal metrics with quasi optimal range of total curvature and precise asymptotic behavior at infinity. If furthermore $K$ is radial symmetric, we establish the same existence result without any growth assumption on $K$.

Theorems & Definitions (22)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Lemma 2.1
• Theorem 2.2
• Theorem 2.3