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

TL;DR

This paper studies the existence and properties of conformal metrics with prescribed Q-curvature in even-dimensional Euclidean spaces, providing necessary conditions and constructing solutions under various growth conditions on the curvature function.

Contribution

It establishes necessary conditions for total curvature and proves existence of solutions with specific asymptotic behaviors under growth constraints on the prescribed curvature.

Findings

Necessary total curvature conditions derived

Existence of solutions with controlled asymptotics shown

Results extended to radially symmetric curvature functions

Abstract

We consider the $Q$-curvature equation \begin{equation}\label{0.1} (-\Delta)^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 $\Lambda_u$ 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 $K$.