Extremal metrics and lower bound of the modified K-energy

TL;DR

This paper offers a new proof that extremal metrics on polarized Kähler manifolds minimize the modified K-energy, utilizing weighted balanced metrics and an idea from C. Li.

Contribution

It introduces a novel proof approach for extremal metrics minimizing the modified K-energy on polarized Kähler manifolds.

Findings

Extremal metrics minimize the modified K-energy.

The proof employs weighted balanced metrics.

The approach adapts C. Li's idea to extremal metrics.

Abstract

We provide a new proof of a result of X.X.Chen and G.Tian : for a polarized extremal K\"ahler manifold, an extremal metric attains the minimum of the modified K-energy. The proof uses an idea of C.Li adapted to the extremal metrics using some weighted balanced metrics.