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

TL;DR

This paper investigates the geometric structure of steady toric K"ahler-Ricci solitons, revealing that under certain conditions, such solitons are necessarily flat, extending classical rigidity results in complex differential geometry.

Contribution

It generalizes Calabi's classical rigidity theorem to steady toric K"ahler-Ricci solitons, showing they must be flat if they have a free torus action.

Findings

The orbit space has a natural Hessian structure with nonnegative Bakry-Émery tensor.

Complete invariant solitons with free torus action are flat.

Extension of classical rigidity results to the steady toric K"ahler setting.

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 $(\mathbb C^*)^n$.