Rigidity of Complete Gradient Shrinkers with Pointwise Pinching Riemannian Curvature

Abstract

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