Entropies in $\mu$-framework of canonical metrics and K-stability, I -- Archimedean aspect: Perelman's W-entropy and $\mu$-cscK metrics

This is the first in a series of two papers (cf. [Ino4]) studying μ-cscK metrics and μK-stability, from a new perspective evoked from observations in [Ino3] and in this first paper. The first paper is about a characterization of μ-cscK metrics in terms of Perelman’s W -entropy W̌λ. We regard Perelman’s W -entropy as a functional on the tangent bundle TH(X,L) of the space H(X,L) of Kähler metrics in a given Kähler class L. The critical points of W̌λ turn out to be μλ-cscK metrics. When λ ≤ 0, the supremum along the fibres gives a smooth functional on H(X,L), which we call μ-entropy. Then μλ-cscK metrics are also characterized as critical points of this functional, similarly as extremal metric is characterized as the critical points of Calabi functional. We also prove theW -entropy is monotonic along geodesics, following Berman– Berndtsson’s subharmonicity argument. Studying the limit of the W -entropy, we obtain a lower bound of the μ-entropy. This bound is not just analogous, but indeed related to Donaldson’s lower bound on Calabi functional by the extremal limit λ→ −∞.