Finite entropy vs finite energy
type: Item
instance of: scholarly article
title: Finite entropy vs finite energy
zbMATH Open document ID: 1473.32014
DOI: 10.4171/CMH/515
author: Eleonora Di Nezza
author: Vincent Guedj
author: Hoang Chinh Lu
published in: Commentarii Mathematici Helvetici
publication date: 2021-07-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/2006.07061
review text: The authors show that the solutions to the degenerate complex Monge-Ampère equation, with the right hand side being a probability measure having bounded entropy, belong to a certain finite energy space. More exactly, let \((X, \omega)\) be a Kähler manifold with volume one and complex dimension \(n\). The authors shows in Theorem A that the solutions \(\phi\) to the equation \[ \omega_\phi^n := (\omega + dd^c \phi)^n = f \omega^n, \] where the probability measure \(f \omega^n\) is assumed to have finite entropy \[ \int_{X} f \log f \omega^n, \] belong to the finite energy class \(\mathcal{E}^p_\omega\) for \(p = \frac{n}{n-1}\) and \(n>1\). The latter space consists of those elements of \(\mathrm{PSH}(X,\omega)\) which satisfy \[ E_p(\phi):=\int_X |\phi|^p \omega_\phi^n < \infty. \] What is more, it is shown that the bound on \(p\) is optimal in Theorem A. On the road for this result the authors provide another interesting result, Theorem B, which is known as the complex Moser-Trudinger inequality. Namely, for the normalized elements \(\phi\) of \(\mathcal{E}^p_\omega\) the inequality \[ \int_X e^{(\mathrm{const} |E_p(\phi)|^{- \frac 1 n}|\phi|^{1+\frac p n})}\omega^n < \mathrm{const} \] is shown to hold.
reviewed by: Marcin Sroka
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 35A23
zbMATH DE Number: 7367627
zbMATH Keywords: Monge-Ampère energy
zbMATH Keywords: entropy
zbMATH Keywords: Moser-Trudinger inequality
MaRDI profile type: Publication
arXiv ID: 2006.07061
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OpenAlex ID: W3173752757
description: scientific article