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.
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.
Keywords: Weak solutions, Hölder continuous, Monge–Ampère, Compact Hermitian manifold
Mot clés : solutions faibles, continue Höldérienne, Monge–Ampère, variété hermitienne compacte
CC-BY-ND 4.0
@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/}
}
TY - JOUR AU - Kołodziej, Sławomir AU - Nguyen, Ngoc Cuong TI - Hölder continuous solutions of the Monge–Ampère equation on compact Hermitian manifolds JO - Annales de l'Institut Fourier PY - 2018 SP - 2951 EP - 2964 VL - 68 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3232/ DO - 10.5802/aif.3232 LA - en ID - AIF_2018__68_7_2951_0 ER -
%0 Journal Article %A Kołodziej, Sławomir %A Nguyen, Ngoc Cuong %T Hölder continuous solutions of the Monge–Ampère equation on compact Hermitian manifolds %J Annales de l'Institut Fourier %D 2018 %P 2951-2964 %V 68 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3232/ %R 10.5802/aif.3232 %G en %F AIF_2018__68_7_2951_0
Kołodziej, Sławomir; Nguyen, Ngoc Cuong. Hölder continuous solutions of the Monge–Ampère equation on compact Hermitian manifolds. Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2951-2964. doi : 10.5802/aif.3232. https://aif.centre-mersenne.org/articles/10.5802/aif.3232/
[1] A new capacity for plurisubharmonic functions, Acta Math., Volume 149 (1982), pp. 1-40 | Zbl
[2] Regularity of plurisubharmonic upper envelopes in big cohomology classes, Perspectives in analysis, geometry, and topology (Stockholm, 2008) (Progress in Mathematics), Volume 296 (2012), pp. 39-66 | Zbl
[3] Pluricomplex energy, Acta Math., Volume 180 (1998) no. 2, pp. 187-217 | Zbl
[4] Hölder regularity for solutions to complex Monge-Ampère equations, Ann. Pol. Math., Volume 113 (2015) no. 2, pp. 109-127 | Zbl
[5] Regularization of closed positive currents of type by the flow of a Chern connection, Contributions to complex analysis and analytic geometry (Paris, 1992) (Aspects of Mathematics), Volume E26, Vieweg, 1994, pp. 105-126 | Zbl
[6] Hölder continuous solutions to Monge-Ampère equations, J. Eur. Math. Soc., Volume 16 (2014) no. 4, pp. 619-647 | Zbl
[7] Pluripotential estimates on compact Hermitian manifolds, Advanced Lectures in Mathematics (ALM), 21, Higher Education Press, 2012, pp. 69-86 | Zbl
[8] Equidistribution speed for Fekete points associated with an ample line bundle, Ann. Sci. Éc. Norm. Supér., Volume 50 (2017) no. 3, pp. 545-578 | Zbl
[9] Characterization of Monge–Ampère measures with Hölder continuous potentials, J. Funct. Anal., Volume 266 (2014) no. 1, pp. 67-84 | Zbl
[10] Exponential estimates for plurisubharmonic functions, J. Differ. Geom., Volume 84 (2010) no. 3, pp. 465-488 | Zbl
[11] Singular Kähler-Einstein metrics, J. Am. Math. Soc., Volume 22 (2009) no. 3, pp. 607-639 | Zbl
[12] Hölder continuous solutions to Monge–Ampère equations, Bull. Lond. Math. Soc., Volume 40 (2008) no. 6, pp. 1070-1080 | Zbl
[13] The Monge–Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J., Volume 52 (2003) no. 3, pp. 667-686 | Zbl
[14] The complex Monge–Ampère equation and pluripotential theory, Mem. Am. Math. Soc., Volume 178 (2005), 64 pages (64 p.) | Zbl
[15] Hölder continuity of solutions to the complex Monge–Ampère equation with the right hand side in : the case of compact Kähler manifolds, Math. Ann., Volume 342 (2008) no. 2, pp. 379-386 | Zbl
[16] Weak solutions to the complex Monge–Ampère equation on Hermitian manifolds, Analysis, complex geometry, and mathematical physics (New York, 2013) (Contemporary Mathematics), Volume 44, American Mathematical Society, 2015, pp. 141-158 | Zbl
[17] Weak solutions of complex Hessian equations on compact Hermitian manifolds, Compos. Math., Volume 152 (2016) no. 11, pp. 2221-2248 | Zbl
[18] Stability and regularity of solutions of the Monge–Ampère equation on Hermitian manifold (2017) (https://arxiv.org/abs/1501.05749)
[19] The complex Monge–Ampère type equation on compact Hermitian manifolds and Applications, Adv. Math., Volume 286 (2016), pp. 240-285 | Zbl
[20] On the Hölder continuous subsolution problem for the complex Monge–Ampèrre equation (2017) (https://arxiv.org/abs/1703.01549)
[21] The complex Monge–Ampère equation on compact Hermitian manifolds, J. Am. Math. Soc., Volume 23 (2010) no. 4, pp. 1187-1195 | Zbl
Cited by Sources:
