A gradient estimate for the Monge-Ampère equation on compact almost Hermitian manifolds
type: Item
instance of: scholarly article
title: A gradient estimate for the Monge-Ampère equation on compact almost Hermitian manifolds
zbMATH Open document ID: 1495.32068
DOI: 10.1215/00192082-9591203
author: Masaya Kawamura
published in: Illinois Journal of Mathematics
publication date: 2022-02-01T00:00:00Z
full work available at URL: 00192082-9591203.full
review text: Let \((X,\omega_0)\) be a closed Kähler manifold. One can formulate a complex Monge-Ampère equation \( (\omega_0 + \frac{i}{2}\partial\overline{\partial}\varphi)^n = e^{f(z)}\omega_0^n\), where \(f(z)\) is a given smooth function satisfying some necessary conditions. The existence of solutions to this equation, famously proven by Yau via the continuity method, establishes the Calabi conjecture and asserts the existence of a unique Ricci-flat Kähler metric when \(X\) has vanishing first Chern class. The work of \textit{J. Chu} et al. [J. Eur. Math. Soc. (JEMS) 21, No. 7, 1949--1984 (2019; Zbl 1452.32035)] establishes a priori estimates when \(X\) is a compact, almost complex manifold. Further work by \textit{B. Guo} et al. [``On \(L^\infty\) estimates for complex Monge-Ampère equations'', Preprint, \url{arXiv:2106.02224}] establishes sharp \(L^\infty\) estimates without pluripotential theory, and uses the Alexandrov-Bakelmann-Pucci (ABP) maximum principle to recover gradient estimates established in [\textit{D. H. Phong} et al., Surv. Differ. Geom. 17, 327--410 (2012; Zbl 1382.32023)]. The paper under review generalizes the above work by establishing gradient estimates in the setting when \((X,J,g)\) is almost Hermitian. That is, when \(g(X,Y) = g(JX,JY)\) for any pair of vector fields \(X, Y\). The paper proves this by again utilizing the ABP maximum principle and explicit computation in local coordinates.
reviewed by: Tobias Shin
Mathematics Subject Classification ID: 32Q60
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 7467755
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: gradient estimates
zbMATH Keywords: almost Hermitian geometry
MaRDI profile type: Publication
OpenAlex ID: W4205270003
cites work: A gradient estimate in the Calabi-Yau theorem
cites work: The Monge-Ampère equation for non-integrable almost complex structures
cites work: Q3084648
cites work: The continuity equation of almost Hermitian metrics
cites work: The Dirichlet problem on almost Hermitian manifolds
cites work: Complex Monge Ampere Equations
cites work: On Hermitian curvature flow on almost complex manifolds
cites work: Nonpositively curved almost Hermitian metrics on product of compact almost complex manifolds
cites work: An almost complex Chern-Ricci flow
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description: scientific article; zbMATH DE number 7467755