Curvature estimates for steady Ricci solitons

TL;DR

This paper establishes exponential decay estimates for the curvature tensor of certain steady Ricci solitons under specific boundedness and decay conditions, advancing understanding of their geometric behavior at infinity.

Contribution

It improves existing curvature decay results for steady Ricci solitons by providing sharper exponential decay estimates under new boundedness and scalar curvature conditions.

Findings

Curvature tensor decays exponentially under bounded potential and curvature conditions.

For 4D steady Ricci solitons with scalar curvature tending to zero, curvature is controlled by scalar curvature.

Exponential decay of curvature occurs when scalar curvature decay is sufficiently fast and potential function is bounded.

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 $|Rm|\leq Ce^{-r}$ 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 $|Rm|\leq cS$ 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.