Modified Extremal Kähler 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 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 $$\sigma $$ implies the existence of a $$\sigma $$ -extremal Kähler metric.