Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. First, if the $Q$-curvature $Q_g\geq 6k^2$ and scalar curvature $R_g\geq -12k$ for some positive constant $k$, then $(M^4,g)$ is either Einstein with $Ric_g=-3kg$ or compact with $R_g\ge 12k$. As a corollary, the fundamental group $\pi_1(M^4)$ satisfies...
Let $(M^n,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$. We show that the existence of a conformal metric with positive $Q$-curvature $Q_g$ and positive scalar curvature $R_g$ is equivalent to the positivity of both the Yamabe invariant $Y(M^n,[g])$ and the Paneitz operator $P_g$. For $n=5$, this eq...
For a smooth, complete and normal metric $g = e^{2u}|dx|^2$ with finite total $n$-th order $Q$-curvature on $\mathbb{R}^n$ with dimension $n \geq 2$, we first show that everywhere non-negativity (resp. non-positivity) $n$-th order $Q$-curvature $Q_g^{(n)}$ implies everywhere non-negativity (resp. non-positivity) of the...
Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the vo...
