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

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 $\lambda \to -\infty$.

v2. 47 pages, Expressions are revised, comments by Laszlo Lempert are added, minimax interpretation of the result is additionally observed, a trailer on the second article was removed. Comments are very welcome!

References in corpus (4)

• On the constant scalar curvature Kähler metrics, apriori estimates
• Geodesic rays and stability in the cscK problem
• Supremum of Perelman's entropy and Kähler-Ricci flow on a Fano manifold
• Equivariant calculus on $\mu$-character and $\mu$K-stability of polarized schemes

Cited by in corpus (1)

• Extremal Kähler metrics on blowups