A gradient estimate in the Calabi-Yau theorem
type: Item
instance of: scholarly article
title: A gradient estimate in the Calabi-Yau theorem
zbMATH Open document ID: 1167.32023
author: Zbigniew Błocki
published in: Mathematische Annalen
publication date: 2009-04-27T00:00:00Z
review text: This paper is a nice contribution to the theory of complex Monge-Ampère equations on compact Kähler manifolds. The setting is the same as in Yau's proof of the Calabi conjecture [\textit{S.-T. Yau}, Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0369.53059)]: \((M,\omega)\) is a compact Kähler manifold and \(\varphi\) is a real function so that \(\omega+i\partial\overline{\partial}\varphi\) is positive definite and with \(\max_M \varphi=0\). One assumes that \[ (\omega+i\partial\overline{\partial}\varphi)^n=f \omega^n, \] for some positive smooth function \(f\) with \(\int_M (f-1)\omega^n=0\), which is the complex Monge-Ampère equation that arises in the Calabi conjecture. In his solution of the conjecture, Yau proved that the function \(\varphi\) has \textit{a priori} bounds in all \(C^k\) norms, depending only on \(M, \omega, f\). To show this, he first proved that the \(C^0\) norm of \(\varphi\) is bounded uniformly, and then used this to derive a \(C^2\) uniform bound. Standard interpolation estimates then provide a \(C^1\) bound on \(\varphi\), but the question remained whether one could derive a \(C^1\) bound on \(\varphi\) depending only on the \(C^0\) bound. The author previously worked on this problem in [Math. Z. 244, No. 1, 153--161 (2003; Zbl 1076.32036)] where he proved such a bound if \((M,\omega)\) has nonnegative bisectional curvature (a rather stringent hypothesis). In this paper, the author generalizes his previous work and proves the desired \(C^1\) bound on \(\varphi\), which depends only on \(M, \omega, f\) and a bound on \(\max_M |\varphi|\). This result has already had some interesting applications, for example by the author as well as D. H. Phong and J. Sturm to the problem of geodesics in the space of Kähler potentials. The author promises further applications to complex Hessian equations.
reviewed by: Valentino Tosatti
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 32Q25
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 5547449
zbMATH Keywords: Calabi-Yau theorem
zbMATH Keywords: gradient estimates
zbMATH Keywords: complex Monge-Ampère equations
MaRDI profile type: Publication
full work available at URL: s00208-008-0307-3
OpenAlex ID: W2135535416
cites work: Q4098875
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DOI: 10.1007/S00208-008-0307-3
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description: scientific article; zbMATH DE number 5547449