On the lower bound of energy functional E_1 (I)-- a stability theorem on the Kaehler Ricci flow

TL;DR

This paper establishes a stability theorem for the Kähler Ricci flow near the infimum of the E_1 energy functional, showing convergence to Kähler Einstein metrics under certain initial conditions.

Contribution

It proves a stability result for the Kähler Ricci flow near the E_1 functional's infimum, advancing understanding of convergence to Kähler Einstein metrics.

Findings

Flow converges to Kähler Einstein metric if initial metric is close and Ricci > -1.

Provides conditions under which the energy functional's lower bound guarantees convergence.

Lays groundwork for proving existence of Kähler Einstein metrics at arbitrary energy levels.

Abstract

In the present paper, we prove a stability theorem for the Kaehler Ricci flow near the infimum of the functional E_1 under the assumption that the initial metric has Ricci > -1 and |Riem| bounded. At present stage, our main theorem still need a topological assumption (1.2) which we hope to be removed in subsequent papers. The underlying moral is: if a Kaehler metric is sufficiently closed to a Kaehler Einstein metric, then the Kaehler Ricci flow converges to it. The present work should be viewed as a first step in a more ambitious program of deriving the existence of Kaehler Einstein metrics with an arbitrary energy level, provided that this energy functional has a uniform lower bound in this Kaehler class.