Steady Ricci solitons with horizontally ϵ-pinched Ricci curvature

Abstract

In this paper, we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally ϵ-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any κ-noncollapsed gradient steady Ricci soliton (Mn, g, f) with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature R(x) limr(x)→∞R(x)f(x) = C0 supx∈MR(x) with satisfies $$C_{0} > \frac{n-2}{2}$$ .