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 $T^n$-invariant shrinking gradient Kähler-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 Kähler-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 $\mathbb{C}$.

arXiv commentaccepted version, to appear in J. London Math. Soc