Modified extremal K\"{a}hler metrics and multiplier Hermitian-Einstein metrics

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