Modified extremal Kähler metrics and multiplier Hermitian-Einstein metrics

Motivated by the notion of multiplier Hermitian-Einstein metric of type $σ$ introduced by Mabuchi, we introduce the notion of $σ$-extremal Kähler metrics on compact Kähler manifolds, which generalizes Calabi's extremal Kähler metrics. We characterize the existence of this metric in terms of the coercivity of a certain functional on the space of Kähler metrics to show that, on a Fano manifold, the existence of a multiplier Hermitian-Einstein metric of type $σ$ implies the existence of a $σ$-extremal Kähler metric.