  We introduce mu-scalar curvature for a K"ahler metric with a moment map mu
and start up a study on constant mu-scalar curvature K"ahler metric as a
generalization of both cscK metric and K"ahler-Ricci soliton and as a
continuity path to extremal metric. We study some fundamental constraints to
the existence of constant mu-scalar curvature K"ahler 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.

Differential Geometry