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 $|\pi_1(M^4)|\leq 16\pi^2/(\int_{M^4}Q_g d\mu_g),$ under the additional assumption $R_g>-12k$. Second, if the scalar curvature $R_g>0$ and $Q_g\geq \theta R_g$ for a positive constant $\theta$, then $M^4$ is compact and the diameter of $(M^4,g)$ is at most $4\pi/\sqrt{15\theta}$.
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 equivalence confirms a conjecture of Gursky-Hang-Lin (2016, IMRN). Furthermore, assuming $Y(M^n,[g])>0$, $Q_g\ge 0$, and $Q_g\not\equiv 0$, we prove that both $R_g$ and $P_g$ are positive which resolves a problem of Hang-Yang (2016, CPAM). As a corollary, we show that the hypotheses of Gursky-Malchiodi (2015, JEMS) are equivalent to those of Hang-Yang (2016, CPAM).
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 sectional curvature. Based on this fact, we secondly show that, once $Q_g^{(n)}$ is non-negative, then for any compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial\Omega$, the following sharp isoperimetric inequality holds: $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}} \left(1 - \frac{2}{(n-1)!\,|\mathbb{S}^n|} \int_{\mathbb{R}^n} Q_g^{(n)} \, d\mu_g\right) |\Omega|_g.$$ The third claim in this article is that, if the $n$-th order $Q$-curvature, $Q_g^{(n)}$, is non-positive and under the main assumption that Cartan-Hadamard conjecture holds true, then we have the sharp inequality $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}}|\Omega|_g.$$
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 volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.