Existence of complete conformal metrics on $\mathbb{R}^n$ with prescribed Q-curvature

TL;DR

This work studies the prescribed Q-curvature problem on $\mathbb{R}^n$ (with $n$ even) by seeking a complete conformal metric $g=e^{2u}|dx|^2$ whose Q-curvature equals a given function $f$ that is positive somewhere and decays like $f(x)=O(|x|^{-l})$ with $l>\frac{n}{2}$. The authors develop a weighted variational framework on a Hilbert space $\mathcal{H}$ and prove a weighted Moser–Trudinger–Adams inequality to control the nonlinear exponential term. A minimization argument yields a smooth solution to $(-\Delta)^{\frac{n}{2}}u=f e^{nu}$, and completeness of the resulting metric follows from Li’s normal-solution criteria for Q-curvature problems. The results establish the existence of complete, finite-total-Q-curvature metrics on $\mathbb{R}^n$ for a sharp decay regime, extending the higher-dimensional theory of Q-curvature with a robust variational approach.

Abstract

Given a smooth function $f(x)$ on $\mathbb{R}^n$ which is positive somewhere and satisfies $f(x)=O(|x|^{-l})$ for any $l>\frac{n}{2}$, we show that there exists a complete and conformal metric $g=e^{2u}|dx|^2$ with finite total Q-curvature such that its Q-curvature equals to $f(x)$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Corollary 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Theorem 3.1
• Corollary 3.1