On convergence of the K\"ahler-Ricci flow

TL;DR

This paper investigates the convergence behavior of the K"ahler-Ricci flow on Fano manifolds, establishing conditions involving eigenvalues and energy decay that ensure convergence.

Contribution

It introduces new convergence criteria for the K"ahler-Ricci flow based on eigenvalue bounds and Mabuchi energy decay, expanding understanding of stability conditions.

Findings

Convergence is achieved under bounded eigenvalues and logarithmic Mabuchi energy decay.

Provides scenarios where Mabuchi energy decay condition is satisfied.

Links spectral and energy conditions to flow stability.

Abstract

We study the convergence of the K\"ahler-Ricci flow on a Fano manifold under some stability conditions. More precisely we assume that the first eingenvalue of the $\bar\partial$-operator acting on vector fields is uniformly bounded along the flow, and in addition the Mabuchi energy decays at most logarithmically. We then give different situations in which the condition on the Mabuchi energy holds.