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=e^2u|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-22. 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.