3d steady Gradient Ricci Solitons with linear curvature decay

Abstract

In this note, we prove that a 3-dimensional steady Ricci soliton is rotationally symmetric if its scalar curvature $R(x)$ satisfies $$\frac{C_0^{-1}}{\rho(x)}\le R(x)\le \frac{C_0}{\rho(x)}$$ for some constant $C_0>0$, where $\rho(x)$ denotes the distance from a fixed point $x_0$. Our result doesn't assume that the soliton is $\kappa$-noncollapsed.