Extremal metrics and lower bound of the modified K-energy
  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. 
Yuji Sano and Carl Tipler
math.DG math.AG