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

TL;DR

This paper introduces a new class of extremal Kähler metrics called σ-extremal metrics, generalizing Calabi's extremal metrics, and links their existence to multiplier Hermitian-Einstein metrics on Fano manifolds.

Contribution

It defines σ-extremal Kähler metrics, characterizes their existence via a functional's coercivity, and connects them to multiplier Hermitian-Einstein metrics on Fano manifolds.

Findings

σ-extremal Kähler metrics generalize Calabi's extremal metrics

Existence characterized by functional coercivity

Multiplier Hermitian-Einstein metrics imply σ-extremal metrics on Fano manifolds

Abstract

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.