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Weighted cscK metric (II): the continuity method

In this paper we investigate the existence of metrics with weighted constant scalar curvature (wcscK for short) on a compact Kähler manifold $X$: this notion include constant scalar curvature Kähler metrics, weighted solitons, Calabi's extremal Kähler metrics and extremal metric on semisimple principal fibrations. We prove that the coercivity of the weighted Mabuchi functional implies the existence of a wcscK metric, thereby achieving the equivalence. \\ We then give several applications in Kähler and toric geometry, such as a weighted version of the toric Yau-Tian-Donaldson correspondence, and the characterization of the existence of wcscK metric on total space of semisimple principal fibration $Y$ in term of existence of wcscK metric on its fiber $X$.