Curvature estimates for steady and expanding solitons in higher dimensions

In this paper, we demonstrate certain curvature estimates on complete non-compact steady and expanding gradient Ricci solitons in higher dimensions. In the expanding case, we prove that if the Ricci curvature decays at least quadratically, then the curvature operator decays at the rate $\text{BigO}(1/r^2)$ when $n=4$ and $\text{BigO}((\log r)/r^2)$ when $n\ge 5$. This refines the curvature bounds in a previous result by Cao-Liu-Xie, and removes the nonnegative Ricci curvature assumption in the estimates by Cao-Liu and Cao-Liu-Xie. As a geometric application, we establish the existence and uniqueness of $C^{1,\alpha }$ conical structure at infinity of Ricci expander with finite Ricci curvature ratio. In the steady case, using an integral estimate of the curvature, we prove that the curvature operator has at most polynomial growth when the potential function is proper and the Ricci curvature has linear decay. Moreover, we also confirm that the curvature is bounded if we further assume the Ricci curvature has super-linear decay $\text{BigO}(r^{-1-\epsilon })$. As an application, we prove the existence and uniqueness of cylindrical structure at infinity of steady soliton with super-linear Ricci curvature decay and proper potential function.