We provide a new proof of a result of X.X. Chen and G.Tian : for a polarized extremal Kähler manifold, the minimum of the modified K-energy is attained at an extremal metric. The proof uses an idea of C. Li  adapted to the extremal metrics using some weighted balanced metrics.
Cite this article
Yuji Sano, Carl Tipler, Extremal metrics and lower bound of the modified K-energy. J. Eur. Math. Soc. 17 (2015), no. 9, pp. 2289–2310DOI 10.4171/JEMS/557