Twisted and conical K\"ahler-Ricci soliton on Fano manifolds

In this paper, we consider the twisted K\"ahler-Ricci soliton, and show that the existence of twisted K\"ahler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical K\"ahler-Ricci soliton, and obtain some existence results. In particular, under some assumptions on the divisor and $\alpha$-invariant, we get the properness of the modified log K-energy and the existence of conical K\"ahler-Ricci soliton with suitable cone angle.