Steady Ricci solitons with horizontally $\epsilon$-pinched Ricci curvature

Abstract

In this paper, we prove that any $\kappa$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $\epsilon$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $\kappa$-noncollapsed gradient steady Ricci soliton $(M^n,g,f)$ with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature $R(x)$ satisfies ${\lim }_{r(x)\to \infty }R(x)f(x)=C_0{\sup }_{x\in M}R(x)$ with $C_0>\frac{n-2}{2}$.

arXiv commentCorollary 3.11 is added