Modified extremal Kähler metrics and multiplier Hermitian-Einstein metrics
Yasuhiro Nakagawa, Satoshi Nakamura
TL;DR
This work develops a variational approach to $\sigma$-extremal Kähler metrics, generalizing Calabi’s extremal metrics through the framework of multiplier Hermitian-Einstein metrics of type $\sigma$ and an $h$-modified extremal equation. By introducing the $h$-modified Mabuchi functional $K_{V}^{h}$ and the $\sigma$-Ding functional $D_{V}^{h}$, the authors prove that existence of a $\sigma$-extremal metric in $\mathcal{H}_{V}$ is equivalent to the coercivity (and invariance) of $K_{V}^{h}$, extending the existence/properness principle of Darvas–Rubinstein. They establish compactness and regularity of minimizers, prove a Bando–Mabuchi type uniqueness, and connect the analytic problem to geodesic stability notions, providing both uniform and standard geodesic stability criteria. Furthermore, they relate $\sigma$-solitons to $\sigma$-extremal metrics, showing existence results on Fano manifolds and, in toric/Fano cases, recovering and extending known soliton–extremal correspondences. The results yield a robust variational and stability framework for canonical metrics with symmetry, linking multiplier Hermitian-Einstein theory to extremal Kähler geometry and its weighted generalizations.
Abstract
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.