Energy functionals and K\"ahler-Ricci solitons

In this paper, we generalize Chen-Tian energy functionals to K"ahler-Ricci solitons and prove that the properness of these functionals is equivalent to the existence of K"ahler-Ricci solitons. We also discuss the equivalence of the lower boundedness of these functionals and their relation with Tian-Zhu’s holomorphic invariant.