Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio

TL;DR

This paper proves that complete gradient expanding Ricci solitons with nonnegative Ricci curvature and finite asymptotic scalar curvature ratio exhibit at least sub-quadratic decay of the Riemann curvature tensor at infinity.

Contribution

It establishes a decay rate for the Riemann curvature tensor in expanding Ricci solitons under finite asymptotic scalar curvature ratio conditions.

Findings

Riemann curvature tensor decays at least sub-quadratically

Finite asymptotic scalar curvature ratio implies specific curvature decay

Results apply to complete gradient expanding Ricci solitons with nonnegative Ricci curvature

Abstract

Let $(M^n, g, f)$, $n\geq 5$, be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature $Rc\geq 0$. In this paper, we show that if the asymptotic scalar curvature ratio of $(M^n, g, f)$ 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 $0<\alpha<2$.