no. 7

Hölder continuous solutions of the Monge–Ampère equation on compact Hermitian manifolds
[Solutions continues Höldériennes de l’équation de Monge–Ampère sur les variétés hermitiennes compactes]
Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2951-2964.

Nous prouvons qu’une mesure de Borel positive avec la masse totale finie, sur une variété hermitienne compacte, admet une solution quasi plurisousharmonique de l’équation de Monge–Ampère si et seulement si elle est dominée localement par des mesures de Monge–Ampère des fonctions plurisousharmoniques continues Höldériennes.

We show that a positive Borel measure of positive finite total mass, on a compact Hermitian manifold, admits a Hölder continuous quasi-plurisubharmonic solution to the Monge–Ampère equation if and only if it is dominated locally by Monge–Ampère measures of Hölder continuous plurisubharmonic functions.

Publié le :
DOI : https://doi.org/10.5802/aif.3232
Classification : 53C55,  35J96,  32U40
Mots clés : solutions faibles, continue Höldérienne, Monge–Ampère, variété hermitienne compacte 
@article{AIF_2018__68_7_2951_0,
     author = {Ko{\l}odziej, S{\l}awomir and Nguyen, Ngoc Cuong},
     title = {H\"older continuous solutions of the {Monge{\textendash}Amp\`ere} equation on compact {Hermitian} manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2951--2964},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {7},
     year = {2018},
     doi = {10.5802/aif.3232},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3232/}
}
Kołodziej, Sławomir; Nguyen, Ngoc Cuong. Hölder continuous solutions of the Monge–Ampère equation on compact Hermitian manifolds. Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2951-2964. doi : 10.5802/aif.3232. https://aif.centre-mersenne.org/articles/10.5802/aif.3232/

[1] Bedford, Eric; Taylor, Bert A. A new capacity for plurisubharmonic functions, Acta Math., Tome 149 (1982), pp. 1-40 | Zbl 0547.32012

[2] Berman, Robert; Demailly, Jean-Pierre Regularity of plurisubharmonic upper envelopes in big cohomology classes, Perspectives in analysis, geometry, and topology (Stockholm, 2008) (Progress in Mathematics) Tome 296 (2012), pp. 39-66 | Zbl 1258.32010

[3] Cegrell, Urban Pluricomplex energy, Acta Math., Tome 180 (1998) no. 2, pp. 187-217 | Zbl 0926.32042

[4] Charabati, Mohamad Hölder regularity for solutions to complex Monge-Ampère equations, Ann. Pol. Math., Tome 113 (2015) no. 2, pp. 109-127 | Zbl 1330.32012

[5] Demailly, Jean-Pierre Regularization of closed positive currents of type (1,1) by the flow of a Chern connection, Contributions to complex analysis and analytic geometry (Paris, 1992) (Aspects of Mathematics) Tome E26, Vieweg, 1994, pp. 105-126 | Zbl 0824.53064

[6] Demailly, Jean-Pierre; Dinew, Sławomir; Guedj, Vincent; Hiep, Pham Hoang; Kołodziej, Sławomir; Zeriahi, Ahmed Hölder continuous solutions to Monge-Ampère equations, J. Eur. Math. Soc., Tome 16 (2014) no. 4, pp. 619-647 | Zbl 1296.32012

[7] Dinew, Sławomir; Kołodziej, Sławomir Pluripotential estimates on compact Hermitian manifolds, Advanced Lectures in Mathematics (ALM), Tome 21, Higher Education Press, 2012, pp. 69-86 | Zbl 1317.32066

[8] Dinh, Tien-Cuong; Ma, Xiaonan; Nguyên, Viêt-Anh Equidistribution speed for Fekete points associated with an ample line bundle, Ann. Sci. Éc. Norm. Supér., Tome 50 (2017) no. 3, pp. 545-578 | Zbl 1379.32016

[9] Dinh, Tien-Cuong; Nguyên, Viêt-Anh Characterization of Monge–Ampère measures with Hölder continuous potentials, J. Funct. Anal., Tome 266 (2014) no. 1, pp. 67-84 | Zbl 1305.32018

[10] Dinh, Tien-Cuong; Nguyên, Viêt-Anh; Sibony, Nessim Exponential estimates for plurisubharmonic functions, J. Differ. Geom., Tome 84 (2010) no. 3, pp. 465-488 | Zbl 1211.32021

[11] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Singular Kähler-Einstein metrics, J. Am. Math. Soc., Tome 22 (2009) no. 3, pp. 607-639 | Zbl 1215.32017

[12] Guedj, Vincent; Kołodziej, Sławomir; Zeriahi, Ahmed Hölder continuous solutions to Monge–Ampère equations, Bull. Lond. Math. Soc., Tome 40 (2008) no. 6, pp. 1070-1080 | Zbl 1157.32033

[13] Kołodziej, Sławomir The Monge–Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J., Tome 52 (2003) no. 3, pp. 667-686 | Zbl 1039.32050

[14] Kołodziej, Sławomir The complex Monge–Ampère equation and pluripotential theory, Mem. Am. Math. Soc., Tome 178 (2005), 64 pages (64 p.) | Zbl 1084.32027

[15] Kołodziej, Sławomir Hölder continuity of solutions to the complex Monge–Ampère equation with the right hand side in L p : the case of compact Kähler manifolds, Math. Ann., Tome 342 (2008) no. 2, pp. 379-386 | Zbl 1149.32018

[16] Kołodziej, Sławomir; Nguyen, Ngoc Cuong Weak solutions to the complex Monge–Ampère equation on Hermitian manifolds, Analysis, complex geometry, and mathematical physics (New York, 2013) (Contemporary Mathematics) Tome 44, American Mathematical Society, 2015, pp. 141-158 | Zbl 1343.32031

[17] Kołodziej, Sławomir; Nguyen, Ngoc Cuong Weak solutions of complex Hessian equations on compact Hermitian manifolds, Compos. Math., Tome 152 (2016) no. 11, pp. 2221-2248 | Zbl 1381.53136

[18] Kołodziej, Sławomir; Nguyen, Ngoc Cuong Stability and regularity of solutions of the Monge–Ampère equation on Hermitian manifold (2017) (https://arxiv.org/abs/1501.05749)

[19] Nguyen, Ngoc Cuong The complex Monge–Ampère type equation on compact Hermitian manifolds and Applications, Adv. Math., Tome 286 (2016), pp. 240-285 | Zbl 1337.53088

[20] Nguyen, Ngoc Cuong On the Hölder continuous subsolution problem for the complex Monge–Ampèrre equation (2017) (https://arxiv.org/abs/1703.01549)

[21] Tosatti, Valentino; Weinkove, Ben The complex Monge–Ampère equation on compact Hermitian manifolds, J. Am. Math. Soc., Tome 23 (2010) no. 4, pp. 1187-1195 | Zbl 1208.53075