On the Geometry of Steady Toric Kähler-Ricci Solitons

In this paper we study gradient steady Kähler-Ricci soliton metrics on non-compact toric manifolds. We show that the orbit space of the free locus of such a manifold carries a natural Hessian structure with a nonnegative Bakry-Émery tensor. We generalize Calabi’s classical rigidity result and use this to prove that any complete \(\varvec{T}^n\) ${\mathit{T}}^n$ -invariant gradient steady Kähler-Ricci soliton with a free torus action must be a flat \((\mathbb {C}^*)^n\) ${({\mathbb{C}}^{\ast })}^n$ .