Classification of 3-dimensional complete rectifiable steady and expanding 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 ${\mathbb{R}}^3$, or (2) the Bryant soliton. In particular, we show that any 3-dimensional complete rectifiable steady gradient Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. Furthermore, we show that any $3$-dimensional complete rectifiable expanding gradient Ricci soliton with positive Ricci curvature is rotationally symmetric.