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

TL;DR

This paper characterizes mu-cscK metrics using Perelman's W-entropy, introduces a related mu-entropy functional, and explores its properties and limits, connecting to K-stability and extremal metrics in Kähler geometry.

Contribution

It provides a new characterization of mu-cscK metrics via Perelman's W-entropy and introduces the mu-entropy functional, linking it to K-stability and extremal metrics.

Findings

W-entropy's critical points are mu-cscK metrics.

W-entropy is monotonic along geodesics.

Lower bounds of mu-entropy relate to Donaldson's bounds.

Abstract

This is the first in a series of two papers studying mu-cscK metrics and muK-stability, from a new perspective evoked from observations in arXiv:2004.06393 and in this first article. The first paper is about a characterization of mu-cscK metrics in terms of Perelman's W-entropy $\check{W}^\lambda$. We regard Perelman's W-entropy as a functional on the tangent bundle $T \mathcal{H} (X, L)$ of the space $\mathcal{H} (X, L)$ of K"ahler metrics in a given K"ahler class $L$. The critical points of $\check{W}^\lambda$ turn out to be $\mu^\lambda$-cscK metrics. When $\lambda \le 0$, the supremum along the fibres gives a smooth functional on $\mathcal{H} (X, L)$, which we call mu-entropy. Then $\mu^\lambda$-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 the W-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 mu-entropy. This bound is not just analogous, but indeed related to Donaldson's lower bound on Calabi functional by the extremal limit $\lambda \to -\infty$.