Continuity method for the Mabuchi soliton on the extremal Fano manifolds

We run the continuity method for Mabuchi's generalization of Kähler-Einstein metrics, assuming the existence of an extremal Kähler metric. It gives an analytic proof (without minimal model program) of the recent existence result obtained by Apostolov, Lahdili and Nitta. Our key observation is the boundedness of the energy functionals along the continuity method. The same argument can be applied to general $g$-solitons and $g$-extremal metrics.