Abstract
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.
Highlights
An n-dimensional ðn ≥ 3Þ Riemannian manifold ðMn, gÞ is called a Ricci soliton if there exist a smooth vector field X and a constant λ ∈ R on Mn such that Rc + L X g= λg, ð1Þ where Rc and LXg denote the Ricci tensor and the Lie derivative of g in the direction of X, respectively, and λ is sometimes called the soliton constant
When X is a gradient of a smooth function f on Mn, the soliton is called a gradient Ricci soliton and (1) becomes
In recent decades, increasing investigations have been done to the rigidity of gradient shrinking Ricci solitons
Summary
An n-dimensional ðn ≥ 3Þ Riemannian manifold ðMn, gÞ is called a Ricci soliton if there exist a smooth vector field X and a constant λ ∈ R on Mn such that. The first rigidity theorem in dimension 3 was proved by Ivey [2] saying that a 3-dimensional compact gradient shrinker is a quotient of S3. When n ≥ 4, under the assumption of nonnegative curvature operator or vanishing Weyl tensor, Naber [4], Ni and Wallach [5], Petersen and Wylie [8], and Zhang [9] proved corresponding rigidity results on gradient shrinkers, which were improved by Catino [10] using a general pointwise pinching condition on the Weyl tensor.
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