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The Eigenvalue Problem for the Complex Monge–Ampère Operator

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Abstract

We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge–Ampère operator on a bounded strongly pseudoconvex domain in \({\mathbb {C}}^n\). We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in \(\Omega \) and boundary values 0. Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P. L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge–Ampère equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of Caffarelli et al. (Commun Pure Appl Math 38(2):209–252, 1985) and Guan (Commun Anal Geom 6(4):687–703, 1998). Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge–Ampère operator.

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Acknowledgements

This article was completed when the Papa Badiane was visiting the Institute of Mathematics of Toulouse during spring 2023. He would like to thank this institution for welcoming him and providing him with excellent research conditions. He also would like to thank especially his coadvisor Salomon Sambou for his strong support and many useful discussion and suggestions during the elaboration of the first part of this work. The Ahmed Zeriahi would like to thank Salomon Sambou and Daouda Niang Diatta for useful mathematical discussions and also for their support and their kind hospitality during his visits to the University Assane Seck of Ziguinchor. The authors would like to thank Vincent Guedj and Chinh H. Lu for interesting discussions and useful suggestions about this work. They are indebted to Chinh Hoang Lu for his careful reading of the first version of this work, pointing out an error and helping them to correct it. Finally, the authors are grateful to the reviewer for his careful reading of the previous version of the paper and for many suggestions that permit to correct some statements and improve significantly the presentation of its final version.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et Applications, Université Assane Seck de Ziguinchor, BP 523, Ziguinchor, Senegal

    Papa Badiane

  2. Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Ahmed Zeriahi

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  1. Papa Badiane
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Correspondence to Ahmed Zeriahi.

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A tribute to Professor Jean-Pierre Demaily.

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This work is dedicated to Professor Jean-Pierre Demailly for his profound contributions to Complex Analysis and Pluripotential Theory with deep applications to Kähler Geometry and complex Algebraic Geometry

The authors were supported by the Direction Europe de la Recherche et Coopération Internationale (DERCI) of CNRS, through the project EMAC.

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Badiane, P., Zeriahi, A. The Eigenvalue Problem for the Complex Monge–Ampère Operator. J Geom Anal 33, 367 (2023). https://doi.org/10.1007/s12220-023-01407-6

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