Modified extremal K\"ahler metrics and multiplier Hermitian-Einstein metrics

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