Entropies in μ-framework of canonical metrics and K-stability, I -- Archimedean aspect: Perelman's W-entropy and μ-cscK metrics · arXivDesk
2101.11197Jan 27, 2021v2. 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!
Entropies in μ-framework of canonical metrics and K-stability, I -- Archimedean aspect: Perelman's W-entropy and μ-cscK metrics
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 Wˇλ. We regard Perelman's W-entropy as a functional on the tangent bundle TH(X,L) of the space H(X,L)
Nearby in the stack
of K"ahler metrics in a given K"ahler class
L
. The critical points of
Wˇλ
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 mu-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 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