Uniqueness of shrinking gradient Kahler-Ricci solitons on non-compact toric manifolds

We show that, up to biholomorphism, there is at most one complete Tn$T<^>n$-invariant shrinking gradient Kahler-Ricci soliton on a non-compact toric manifold M. We also establish uniqueness without assuming Tn$T<^>n$-invariance if the Ricci curvature is bounded and if the soliton vector field lies in the Lie algebra t$\mathfrak {t}$ of Tn$T<^>n$. As an application, we show that, up to isometry, the unique complete shrinking gradient Kahler-Ricci soliton with bounded scalar curvature on CP1xC$\mathbb {C}\mathbb {P}<^>{1} \times \mathbb {C}$ is the standard product metric associated to the Fubini-Study metric on CP1$\mathbb {C}\mathbb {P}<^>{1}$ and the Euclidean metric on C$\mathbb {C}$.