On geometry of steady toric Kähler-Ricci solitons

Abstract

In this paper we study the gradient steady Kähler-Ricci soliton metrics on noncompact 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 T n -invariant gradient steady Kähler-Ricci soliton with a free torus action must be a flat (C * ) n .