On geometry of Q⁽²ᵏ⁾g-curvature

The main purpose of current article is to study the geometry of Q-curvature. For simplicity, we start with a very simple model: a complete and conformal metric g=e²ᵘ|dx|² on Rⁿ. We assume the metric g has non-negative $nth$-order Q-curvature and non-negative scalar curvature. We show that the Ricci curvature is non-negative. We further assume that the metric g is not degenerate in the sense that the isoperimetric ratio near the end is positive. With this extra assumption, we show that the growth rate of $kth$ elementary symmetric function σₖ(g) of Ricci curvature over geodesic ball of radius r is at most polynomial in r with order $n-2k$ for all 1 ≤ k ≤ n-2/2. Similarly, we are able to show that the same growth control holds for $2kth$-order Q-curvature. Finally, we also observe that for $k=1$ or $2$, the gap theorems for Q⁽²ᵏ⁾g hold true.