Uniformly strong convergence of K\"ahler-Ricci flows on a Fano manifold

TL;DR

This paper proves the uniform strong convergence of K"ahler-Ricci flows on Fano manifolds with varied initial conditions, establishing the uniqueness of K"ahler-Ricci solitons and generalizing previous results.

Contribution

It demonstrates the uniform strong convergence of K"ahler-Ricci flows on Fano manifolds with varied initial metrics and structures, extending existing uniqueness theorems.

Findings

Proves uniform strong convergence of K"ahler-Ricci flows.

Establishes the uniqueness of K"ahler-Ricci solitons.

Generalizes previous theorems on K"ahler-Einstein metrics.

Abstract

In this paper, we study the uniformly strong convergence of K\"ahler-Ricci flow on a Fano manifold with varied initial metrics and smooth deformation complex structures. As an application, we prove the uniqueness of K\"ahler-Ricci solitons in sense of diffeomorphism orbits. The result generalizes Tian-Zhu's theorem for the uniqueness of of K\"ahler-Ricci solitons on a compact complex manifold, and it is also a generalization of Chen-Sun's result of for the uniqueness of of K\"ahler-Einstein metric orbits.