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Gradient and Hessian Estimates for the Hermitian Monge–Ampère Equation

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Abstract

We substantially improve our previous gradient estimate for the Monge–Ampère equation on a compact Hermitian manifold. The improvements concern the structure assumptions on the second member. We also give an estimate for the non-mixed second-order derivatives. These estimates are required to apply either the Evans–Krylov estimates or the third derivatives estimates for equations with a gradient term.

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Acknowledgements

The author wishes to express his gratitude to the referee for bringing to his attention some recent papers on the topic and for pointing out some inaccuracies in the previous version of this paper. His/her remarks improves significantly the presentation.

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  1. Mathématiques-Bâtiment M2, Université de Lille, 59655, Villeneuve d’Ascq Cedex, France

    Abdellah Hanani

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  1. Abdellah Hanani
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I wrote the main manuscript text and revised the manuscript.

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Correspondence to Abdellah Hanani.

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Hanani, A. Gradient and Hessian Estimates for the Hermitian Monge–Ampère Equation. Mediterr. J. Math. 20, 291 (2023). https://doi.org/10.1007/s00009-023-02493-0

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  • DOI: https://doi.org/10.1007/s00009-023-02493-0

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