Curvature estimates for steady Ricci solitons

Abstract

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 ${\overline{\lim }}_{r\to \infty }r|Rm|<\frac{1}{5}$, then $|Rm|\le C{e}^{-r}$ for some constant $C>0$, improving a result of [36]. 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 $|Rm|\le cS$ for some constant $c>0$, improving an estimate in [11]. As an application, we show that for a four dimensional complete non Ricci flat gradient steady Ricci soliton, $|Rm|$ decays exponentially provided that ${\overline{\lim }}_{r\to \infty }rS$ is sufficiently small and $f$ is bounded above by a constant.

References in corpus (2)

• Geometry of shrinking Ricci solitons
• Classification of expanding and steady Ricci solitons with integral curvature decay