A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

Abstract

Let $(M^n, g, f)$ be an $n$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$. 1. If its scalar curvature is $\frac{k}{2}$, Ricci curvature is nonnegative and sectional curvature has upper bound $\frac{1}{2(k-1)}$, we prove that the Ricci shrinker is isometric to a finite quotient of $\mathbb{R}^{n-k}\times \mathbb{S}^k$. 2. If $M$ has constant scalar curvature $R=\frac{n-2}{2}$, and each level set of $f$ has vanishing Weyl curvature, we prove that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^{n-2}$. This can be seen a generalization of Cheng-Zhou's four dimensional result \cite{Cheng-Zhou} to high dimension, since the level set of the potential function $f$ has vanishing Weyl curvature automatically when $n=4$.