Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio

Let ( M n , g , f ) , n ≥ 5 , be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature R c ≥ 0 . In this paper, we show that if the asymptotic scalar curvature ratio of ( M n , g , f ) is finite (i.e., lim sup r → ∞ R r 2 < ∞ ), then the Riemann curvature tensor must have at least sub-quadratic decay, namely, lim sup r → ∞ | R m | r α < ∞ for any 0 < α < 2 .