Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio
Let \((M^n, g, f)\), \(n\ge 5\), be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature \(Rc\geq 0\). We show that if in addition the asymptotic scalar curvature ratio is finite (i.e., \(\limsup*{r\to \infty} R r^2< \infty\)), then the Riemann curvature tensor must have at least sub-quadratic decay, namely, \(\limsup{r\to \infty}\) \(\|Rm\| r^{\alpha}< \infty\) for any \(0<\alpha<2\).
