K\"ahler-Ricci flow on a toric manifold with positive first Chern class

TL;DR

This paper proves that on a compact toric manifold with positive first Chern class, the Kähler-Ricci flow starting from any invariant metric converges to a Kähler-Ricci soliton, providing an alternative proof of their existence.

Contribution

It demonstrates convergence of the Kähler-Ricci flow to solitons on toric manifolds with positive first Chern class, offering a new proof of soliton existence.

Findings

Flow converges to a Kähler-Ricci soliton

Provides an alternative proof for soliton existence

Applicable to any initial invariant metric

Abstract

In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the K\"ahler-Ricci flow with any initial $(S^1)^n$-invariant K\"ahler metric converges to a K\"ahler-Ricci soliton. In particular, we give another proof for the existence of K\"ahler-Ricci solitons on a compact toric manifold with positive first Chern class by using the K\"ahler-Ricci flow.