Rigidity of Complete Gradient Shrinkers with Pointwise Pinching Riemannian Curvature

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