Curvature estimates for steady Ricci solitons

We show that for an $n$ dimensional complete non-Ricci flat gradient steady Ricci soliton with potential function $f$ bounded above by a constant and curvature tensor $Rm$ satisfying $\overline {\lim }_{r\to \infty } r\vert Rm\vert<\frac {1}{5}$, $\vert Rm\vert\leq Ce^{-r}$ for some constant $C>0$, improving on a result of Munteanu, Sung, and Wang's. For any four dimensional complete non-Ricci flat gradient steady Ricci soliton with scalar curvature $S\to 0$ as $r\to \infty$, we prove that $\vert Rm\vert\leq cS$ for some constant $c>0$, improving on an estimate by Cao and Cui. As an application, we show that for a four dimensional complete non-Ricci flat gradient steady Ricci soliton, $\vert Rm\vert$ decays exponentially provided that $\overline {\lim }_{r\to \infty } rS$ is sufficiently small and $f$ is bounded above by a constant.