Classification of 3-dimensional complete rectifiable steady gradient Ricci solitons

Let $(M,g,f)$ be a 3-dimensional complete steady gradient Ricci soliton. Assume that $M$ is rectifiable, that is, the potential function can be written as $f=f(r)$, where $r$ is a distance function. Then, we prove that $M$ is isometric to (1) a quotient of $\mathbb{R}^3$, (2) a quotient of $\Sigma^2\times \mathbb{R}$, where $\Sigma^2$ is the Hamilton's cigar soliton, or (3) the Bryant soliton. In particular, we show that any 3-dimensional complete rectifiable steady gradient Ricci soliton with positive sectional curvature is isometric to the Bryant soliton.Comment: 7 page