Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds

Lu, Chinh H.; Phung, Trong-Thuc; Tô, Tât-Dat

doi:10.5802/aif.3436

Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds
[Stabilité et Continuité Höldérienne des solutions des équations de Monge–Ampère complexes sur des variétés Hermitiennes compactes]

Lu, Chinh H.¹ ; Phung, Trong-Thuc ² ; Tô, Tât-Dat ^3,⁴

¹ Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405, Orsay, (France)
² Ho Chi Minh City University of Technology, VNU-HCM, (Vietnam)
³ Current address: Institut de Mathématiques de Jussieu-Paris Rive Gauche Sorbonne Université - Campus Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05 (France)
⁴ École Nationale de l’Aviation Civile Université de Toulouse 7, Avenue Edouard Belin FR-31055 Toulouse Cedex 04, (France)

Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 2019-2045.

Résumé
Abstract

Soit $(X, ω)$ une variété Hermitienne compacte de dimension $n$ . On établit la stabilité des solutions des équations de Monge–Ampère avec second membre dans $L^{p}$ , $p > 1$ . En utilisant ce résultat on montre que les solutions sont continues höldériennes avec le même exposant que celui obtenu dans le cas Kählérien par Demailly-Dinew–Guedj–Kołodziej–Pham–Zeriahi. Notre méthode s’applique également dans le contexte des classes de cohomologie sur une variété Kählérienne.

Let $(X, ω)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge–Ampère equations with right-hand side in $L^{p}$ , $p > 1$ . Using this we prove that the solutions are Hölder continuous with the same exponent as in the Kähler case by Demailly–Dinew–Guedj–Kołodziej–Pham–Zeriahi. Our techniques also apply to the setting of big cohomology classes on compact Kähler manifolds.

Reçu le : 2020-03-24
Révisé le : 2020-08-08
Accepté le : 2020-11-13
Première publication : 2021-12-15
Publié le : 2022-03-15

DOI : 10.5802/aif.3436

Classification : 32W20, 32U05, 32Q15
Keywords: Hermitian manifold, Complex Monge–Ampère equation, Stability, Comparison principle
Mot clés : Variété hermitienne, Équation de Monge–Ampère complexe, Stabilité, Principe de comparaison

Affiliations des auteurs :

Lu, Chinh H. ¹ ; Phung, Trong-Thuc ² ; Tô, Tât-Dat ^{3, 4}

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Lu, Chinh H. and Phung, Trong-Thuc and T\^o, T\^at-Dat},
     title = {Stability and {H\"older} regularity of solutions to complex {Monge{\textendash}Amp\`ere} equations on compact {Hermitian} manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2019--2045},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {5},
     year = {2021},
     doi = {10.5802/aif.3436},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3436/}
}

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Lu, Chinh H.; Phung, Trong-Thuc; Tô, Tât-Dat. Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds. Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 2019-2045. doi : 10.5802/aif.3436. https://aif.centre-mersenne.org/articles/10.5802/aif.3436/

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