K\"ahler Ricci flow with vanished Futaki invariant

TL;DR

This paper investigates the convergence and stability of the K"ahler-Ricci flow on compact K"ahler manifolds with positive first Chern class and zero Futaki invariant, providing criteria for stability near K"ahler-Einstein metrics.

Contribution

It establishes a new stability criterion for the K"ahler-Ricci flow around K"ahler-Einstein metrics with positive scalar curvature, even with nontrivial holomorphic vector fields.

Findings

Convergence of K"ahler-Ricci flow under specific conditions.

A stability theorem for K"ahler-Einstein metrics with nontrivial holomorphic vector fields.

Criteria for stability with perturbed complex structures.

Abstract

We study the convergence of the K\"ahler-Ricci flow on a compact K\"ahler manifold $(M,J)$ with positive first Chern class $c_1(M;J)$ and vanished Futaki invariant on $\pi c_1(M;J)$. As the application we establish a criterion for the stability of the K\"ahler-Ricci flow (with perturbed complex structure) around a K\"ahler-Einstein metric with positive scalar curvature, under certain local stable condition on the dimension of holomorphic vector fields. In particular this gives a stability theorem for the existence of K\"ahler-Einstein metrics on a K\"ahler manifold with possibly nontrivial holomorphic vector fields.