Uniqueness of shrinking gradient Kähler–Ricci solitons on non‐compact toric manifolds

Abstract

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