Rigidity of shrinking gradient ricci soliton with constant scalar curvature

Abstract

Let $(M^n,g,f)$ be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that (i) $Ric\ge \frac{{\nabla }_{\nabla f}Ric}{f}$ on $M\setminus D$, where $D$ is a compact set over $M$; (ii) $(M^n,g,f)$ smoothly converges to ${\mathbb{R}}^2\times {\mathbb{S}}^{n-2}$, we conclude that $(M^n,g,f)$ is isometric to ${\mathbb{R}}^2\times {\mathbb{S}}^{n-2}$. Notably, condition \textup{(i)} is weaker than the radial flatness condition in \cite{Petersen-Wylie2}.