We show that the Bryant soliton is the unique asymptotically cylindrical steady gradient Ricci soliton, in any dimension $n \geq 3$ and without any curvature assumptions. This generalizes a celebrated theorem of Brendle. We also prove that any steady gradient Ricci soliton asymptotic to a cylinder over the homogeneous lens space $\mathbb{ S}^{ 2m+1}/\mathbb{ Z}_k = L_{ m,k}$, for $m \geq 1$ and $k \geq 3$, is a noncollapsed Appleton soliton on the complex line bundle $O(-k)$ over $\mathbb{ CP}^m$. Specializing to dimension 4, we classify steady gradient Ricci soliton singularity models on smooth orbifolds with tangent flows at infinity of the form $(SU(2)/\Gamma) \times \mathbb{ R}$.
Categories: math.DG