Rigidity of complete gradient shrinkers with pointwise pinching Riemannian curvature (Q2051974)

Summary: Let \((M^n, g, f)\) be a complete gradient shrinking Ricci soliton of dimension \(n\geq3\). 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 \(\mathbb{R}^n\) 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 \((M^n, g, f)\), such as the property of \(f\)-parabolic and a Liouville type theorem.