Constant $$\mu $$-Scalar Curvature Kähler Metric—Formulation and Foundational Results

Abstract

We introduce \(\mu \)-scalar curvature for a Kähler metric with a moment map \(\mu \) and start up a study on constant \(\mu \)-scalar curvature Kähler metric as a generalization of both cscK metric and Kähler–Ricci soliton and as a continuity path to extremal metric. We study some fundamental constraints to the existence of constant \(\mu \)-scalar curvature Kähler metric by investigating a volume functional as a generalization of Tian-Zhu’s work, which is closely related to Perelman’s W-functional. A new K-energy is studied as an approach to the uniqueness problem of constant \(\mu \)-scalar curvature and as a prelude to new K-stability concept.