On geometry of steady toric K\"ahler-Ricci solitons

Abstract

In this paper we study the gradient steady K\"ahler-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-\'Emery tensor. We generalize Calabi's classical rigidity result and use this to prove that any complete $\mathbf T^n$-invariant gradient steady K\"ahler-Ricci soliton with a free torus action must be a flat $(\mathbb C^*)^n$.