Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio

Abstract

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

20 pages. Lemma 3.1 statement simplified; the proof of Lemma 3.2 & Lemma 3.3 streamlined. arXiv admin note: text overlap with arXiv:2111.09848

References in corpus (1)

• Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

Cited by in corpus (1)

• Curvature estimates for four-dimensional complete gradient expanding Ricci solitons