A note on diastatic entropy and balanced metrics

TL;DR

This paper investigates the relationship between diastatic entropy and balanced Kähler metrics on complex bounded domains, providing bounds, characterizations, and explicit computations in special cases.

Contribution

It establishes an upper bound for diastatic entropy based on the balanced condition and proves the converse for homogeneous domains, including explicit entropy calculations.

Findings

Upper bound for diastatic entropy in terms of balanced metrics

Characterization of balanced metrics via entropy bounds in homogeneous domains

Explicit computation of entropy using Piatetski-Shapiro constants

Abstract

We give un upper bound Ent(\Omega, g)<\lambda\ of the diastatic entropy Ent(\Omega, g) of a complex bounded domain (\Omega, g) in terms of the balanced condition (in Donaldson terminology) of the Kaehler metric \lambda g. When (\Omega, g) is a homogeneous bounded domain we show that the converse holds true, namely if Ent(\Omega, g)<1 then g is balanced. Moreover, we explcit compute Ent(\Omega, g) in terms of Piatetski-Shapiro constants.