Rigidity of Complete Gradient Shrinkers with Pointwise Pinching Riemannian Curvature

Abstract

Let $\left(M^n,g,f\right)$ be a complete gradient shrinking Ricci soliton of dimension $n\ge 3$ . In this paper, we study the rigidity of $\left(M^n,g,f\right)$ with pointwise pinching curvature and obtain some rigidity results. In particular, we prove that every $n$ -dimensional gradient shrinking Ricci soliton $\left(M^n,g,f\right)$ is isometric to ${ℝ}^n$ or a finite quotient of ${\mathbb{S}}^n$ under some pointwise pinching curvature condition. The arguments mainly rely on algebraic curvature estimates and several analysis tools on $\left(M^n,g,f\right)$ , such as the property of $f$ -parabolic and a Liouville type theorem.