The modified K\"ahler-Ricci flow and solitons
  We investigate the K\"ahler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a K\"ahler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy. 
D.H. Phong, Jian Song, Jacob Sturm and Ben Weinkove