Some new estimates for the complex Monge-Ampère equation
type: Item
instance of: scholarly article
title: Some new estimates for the complex Monge-Ampère equation
zbMATH Open document ID: 1435.32051
author: X.-X. Chen
author: Jingrui Cheng
published in: Science China. Mathematics
publication date: 2019-11-27T00:00:00Z
review text: The authors study regularity estimates for solutions of complex Monge-Ampère equations on an \(n\)-dimensional compact Kähler manifold \((X, \omega)\). More precisely, they look at equations of the type \[ (\omega+i\partial\bar{\partial} \varphi )^n= e^F \omega^n. \] The question is then to understand the regularity of \(\varphi\) depending on the regularity of \(F\). Several results are already known. Among these: if \(F\in C^\infty\), then \(\varphi\in C^\infty\) [\textit{S.-T. Yau}, Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0369.53059)]; if \(e^F\in L^p\), \(p>1\) then \(\varphi\in C^\alpha\), for some \(\alpha\in(0,1)\) [\textit{S. Kołodziej}, Acta Math. 180, No. 1, 69--117 (1998; Zbl 0913.35043)]; if \(F\) is Lipschitz, then there is a bound on \(\nabla \varphi\) (given a bound on \( \|\varphi\|_{C^0}\)) [\textit{Z. Błocki}, Math. Ann. 344, No. 2, 317--327 (2009; Zbl 1167.32023)]. The paper makes a contribution in this direction: the authors prove that if \(F\) is merely continuous then \(\nabla \varphi\in L^p\) for all \(p\in (0, +\infty)\), giving a very precise bound on the \(L^p\)-norm of \(\nabla \varphi\) depending on \(\omega\), on \(|F|\) and on the \textit{modulus of continuity} of \(F\). Moreover, when \(n=2\) they are able to get rid of the dependence on the latter quantity. It is worth observing that this (integral) gradient estimate does not require any bound on the derivatives of \(F\). As a corollary they also obtain pointwise gradient estimates: if \(F\in C^\alpha\) for some \(\alpha\in(0,1)\), then \(|\nabla \varphi|<C \) pointwise. The authors also generalise all of the above results to the case of manifolds with boundary (this is done in Section 3). In the last Section they show an integrability estimate for \(e^{-\varphi}\) assuming an extra condition on \(F\) (e.g., \(F\) with \textit{finite entropy}). The latter result can be obtained as a corollary of some key facts in pluripotential theory (that hold under much weaker conditions o…
reviewed by: Eleonora Di Nezza
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 35R01
zbMATH DE Number: 7137523
zbMATH Keywords: Kähler manifolds
zbMATH Keywords: Monge-Ampère equations
MaRDI profile type: Publication
full work available at URL: s11425-019-1600-8
OpenAlex ID: W2980475809
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DOI: 10.1007/S11425-019-1600-8
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description: scientific article; zbMATH DE number 7137523