Uniqueness of shrinking gradient K\"ahler-Ricci solitons on non-compact toric manifolds

We show that, up to biholomorphism, there is at most one complete $T^n$-invariant shrinking gradient K\"ahler-Ricci soliton on a non-compact toric manifold $M$. We also establish uniqueness without assuming $T^n$-invariance if the Ricci curvature is bounded and if the soliton vector field lies in the Lie algebra $\mathfrak{t}$ of $T^n$. As an application, we show that, up to isometry, the unique complete shrinking gradient K\"ahler-Ricci soliton with bounded scalar curvature on $\mathbb{CP}^{1} \times \mathbb{C}$ is the standard product metric associated to the Fubini-Study metric on $\mathbb{CP}^{1}$ and the Euclidean metric on $\mathbb{C}$.