Uniqueness of asymptotically cylindrical steady gradient Ricci solitons

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 $\mathbb{CP}^m$. In dimension 4, our results lead to a classification of steady gradient Ricci soliton singularity models on smooth manifolds which possess a tangent flow at infinity of the form $(SU(2)/\Gamma) \times \mathbb{R}$.