Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds
Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2019-2045.

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.

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.

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DOI: 10.5802/aif.3436
Classification: 32W20,  32U05,  32Q15
Keywords: Hermitian manifold, Complex Monge–Ampère equation, Stability, Comparison principle
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)
License: CC-BY-ND 4.0
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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, Volume 71 (2021) no. 5, pp. 2019-2045. doi : 10.5802/aif.3436. https://aif.centre-mersenne.org/articles/10.5802/aif.3436/

[1] Bedford, Eric; Taylor, B. A. The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math., Volume 37 (1976) no. 1, pp. 1-44 | DOI | Zbl

[2] Berman, Robert J.; Boucksom, Sébastien; Guedj, Vincent; Zeriahi, Ahmed A variational approach to complex Monge–Ampère equations, Publ. Math., Inst. Hautes Étud. Sci., Volume 117 (2013), pp. 179-245 | DOI | Numdam | Zbl

[3] Berman, Robert J.; Demailly, Jean-Pierre Regularity of plurisubharmonic upper envelopes in big cohomology classes, Perspectives in analysis, geometry, and topology. On the occasion of the 60th birthday of Oleg Viro. Based on the Marcus Wallenberg symposium on perspectives in analysis, geometry, and topology, Stockholm, Sweden, May 19–25, 2008 (Progress in Mathematics), Volume 296, Birkhäuser, 2012, pp. 39-66 | DOI | MR | Zbl

[4] Błocki, Zbigniew On uniform estimate in Calabi–Yau theorem, Sci. China, Ser. A, Volume 48 (2005) no. suppl., pp. 244-247 | DOI | MR | Zbl

[5] Błocki, Zbigniew On the uniform estimate in the Calabi–Yau theorem. II, Sci. China, Math., Volume 54 (2011) no. 7, pp. 1375-1377 | DOI | MR | Zbl

[6] Bloom, Thomas; Levenberg, Norman Pluripotential energy, Potential Anal., Volume 36 (2012) no. 1, pp. 155-176 | DOI | MR | Zbl

[7] Boucksom, Sébastien Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 1, pp. 45-76 | DOI | Numdam | MR | Zbl

[8] Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | DOI | Zbl

[9] Chen, Xiuxiong On the existence of constant scalar curvature Kähler metric: a new perspective (2015) (https://arxiv.org/abs/1506.06423 to appear in theAnnales mathématiques de Québec)

[10] Chen, Xiuxiong; Cheng, Jingrui On the constant scalar curvature Kähler metrics, apriori estimates (2017) (https://arxiv.org/abs/1712.06697)

[11] Chen, Xiuxiong; Cheng, Jingrui On the constant scalar curvature Kähler metrics, existence results (2018) (https://arxiv.org/abs/1801.00656)

[12] Chen, Xiuxiong; Cheng, Jingrui On the constant scalar curvature Kähler metrics, general automorphism group (2018) (https://arxiv.org/abs/1801.05907)

[13] Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler–Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 183-197 | DOI | Zbl

[14] Cherrier, Pascal Équations de Monge–Ampère sur les variétés Hermitiennes compactes, Bull. Sci. Math., Volume 111 (1987), pp. 343-385 | MR | Zbl

[15] Chiose, Ionut On the invariance of the total Monge–Ampère volume of Hermitian metrics (2016) (https://arxiv.org/abs/1609.05945)

[16] Chu, Jianchu; Zhou, Bin Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds, Sci. China, Math., Volume 62 (2019) no. 2, pp. 371-380 | DOI | MR | Zbl

[17] Darvas, Támas Geometric pluripotential theory on Kähler manifolds, Advances in complex geometry. Contributions from the JHU-UMD complex geometry seminar, John Hopkins University, Baltimore, MD, USA and University of Maryland, College Park, MD, USA, 2015–2018 (Contemporary Mathematics), Volume 735, American Mathematical Society, 2019, pp. 1-104 | DOI | Zbl

[18] Darvas, Támas; Di Nezza, Eleonora; Lu, Chinh H. L 1 metric geometry of big cohomology classes, Ann. Inst. Fourier, Volume 68 (2018) no. 7, pp. 3053-3086 | DOI | Numdam | MR | Zbl

[19] Darvas, Támas; Di Nezza, Eleonora; Lu, Chinh H. Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity, Anal. PDE, Volume 11 (2018) no. 8, pp. 2049-2087 | DOI | Zbl

[20] Darvas, Támas; Di Nezza, Eleonora; Lu, Chinh H. On the singularity type of full mass currents in big cohomology classes, Compos. Math., Volume 154 (2018) no. 2, pp. 380-409 | DOI | MR | Zbl

[21] Darvas, Támas; Di Nezza, Eleonora; Lu, Chinh H. Log-concavity of volume and complex Monge–Ampère equations with prescribed singularity (2019) (https://arxiv.org/abs/1807.00276, to appear in Mathematische Annalen)

[22] Darvas, Támas; Di Nezza, Eleonora; Lu, Chinh H. The metric geometry of singularity types (2019) https://arxiv.org/abs/1909.00839, to appear in Journal für die Reine und Angewandte Mathematik (Crelle’s Journal) | DOI

[23] Demailly, Jean-Pierre Regularization of closed positive currents and intersection theory, J. Algebr. Geom., Volume 1 (1992) no. 3, pp. 361-409 | MR | Zbl

[24] 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. Based on a colloquium dedicated to Pierre Dolbeault, Paris, France, June 23-26, 1992 (Aspects of Mathematics), Volume E26, Vieweg & Sohn, 1994, pp. 105-126 | Zbl

[25] Demailly, Jean-Pierre Complex Analytic and Differential Geometry, 2012 (Demailly’s webpage: https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)

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

[27] Dinew, Sławomir An inequality for mixed Monge–Ampère measures, Math. Z., Volume 262 (2009) no. 1, pp. 1-15 | DOI | Zbl

[28] Dinew, Sławomir Uniqueness in (X,ω), J. Funct. Anal., Volume 256 (2009) no. 7, pp. 2113-2122 | DOI | Zbl

[29] Dinew, Sławomir Pluripotential theory on compact Hermitian manifolds, Ann. Fac. Sci. Toulouse, Math., Volume 25 (2016) no. 1, pp. 91-139 | DOI | Numdam | MR | Zbl

[30] Dinew, Sławomir; Kołodziej, Sławomir Pluripotential estimates on compact Hermitian manifolds, Advances in geometric analysis. Collected papers of the workshop on geometry in honour of Shing-Tung Yau’s 60th birthday, Warsaw, Poland, April 6–8, 2009 (Advanced Lectures in Mathematics (ALM)), Volume 21, International Press, 2012, pp. 69-86 | MR | Zbl

[31] Dinew, Sławomir; Zhang, Zhou On stability and continuity of bounded solutions of degenerate complex Monge–Ampère equations over compact Kähler manifolds, Adv. Math., Volume 225 (2010) no. 1, pp. 367-388 | DOI | Zbl

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

[33] Dinh, Tien-Cuong; Nguyên, Viêt-Anh; Sibony, Nessim Exponential estimates for plurisubharmonic functions and stochastic dynamics, J. Differ. Geom., Volume 84 (2010) no. 3, pp. 465-488 | MR | Zbl

[34] Dinh, Tien-Cuong; Sibony, Nessim Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms, J. Algebr. Geom., Volume 19 (2010) no. 3, pp. 473-529 | DOI | Zbl

[35] Gauduchon, Paul Le théorème de l’excentricité nulle, C. R. Math. Acad. Sci. Paris, Volume 285 (1977) no. 5, pp. 387-390 | Zbl

[36] Guan, Bo; Li, Qun Complex Monge–Ampère equations and totally real submanifolds, Adv. Math., Volume 225 (2010) no. 3, pp. 1185-1223 | DOI | Zbl

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

[38] Guedj, Vincent; Lu, Chinh H.; Zeriahi, Ahmed Stability of solutions to complex Monge–Ampère flows, Ann. Inst. Fourier, Volume 68 (2018) no. 7, pp. 2819-2836 | DOI | Numdam | Zbl

[39] Guedj, Vincent; Zeriahi, Ahmed Stability of solutions to complex Monge–Ampère equations in big cohomology classes, Math. Res. Lett., Volume 19 (2012) no. 5, pp. 1025-1042 | DOI | Zbl

[40] Guedj, Vincent; Zeriahi, Ahmed Degenerate complex Monge–Ampère equations, EMS Tracts in Mathematics, 26, European Mathematical Society, 2017 | DOI | Zbl

[41] Kołodziej, Sławomir Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge–Ampère operator, Ann. Pol. Math., Volume 65 (1996) no. 1, pp. 11-21 | DOI | Zbl

[42] Kołodziej, Sławomir The complex Monge–Ampère equation, Acta Math., Volume 180 (1998) no. 1, pp. 69-117 | DOI | Zbl

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

[44] 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., Volume 342 (2008) no. 2, pp. 379-386 | DOI | Zbl

[45] Kołodziej, Sławomir Weak solutions of the Monge–Ampère equation on compact Hermitian manifolds, Int. J. Math., Volume 28 (2017) no. 9, 1740002 | DOI | Zbl

[46] Kołodziej, Sławomir; Cuong, Nguyen Ngoc Weak solutions to the complex Monge–Ampère equation on Hermitian manifolds, Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong. Proceedings of the conference, Columbia University, New York, NY, USA, May 7–11, 2013 (Contemporary Mathematics), Volume 644, American Mathematical Society, 2015, pp. 141-158 | DOI | Zbl

[47] Kołodziej, Sławomir; Cuong, Nguyen Ngoc Hölder continuous solutions of the Monge–Ampère equation on compact Hermitian manifolds, Ann. Inst. Fourier, Volume 68 (2018) no. 7, pp. 2951-2964 | DOI | Numdam | Zbl

[48] Kołodziej, Sławomir; Cuong, Nguyen Ngoc Stability and regularity of solutions of the Monge–Ampère equation on Hermitian manifolds, Adv. Math., Volume 346 (2019), pp. 264-304 | DOI | Zbl

[49] Kołodziej, Sławomir; Cuong, Nguyen Ngoc Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity (2020) (https://arxiv.org/abs/2003.05061)

[50] Lu, Chinh H.; Nguyên, Van-Dong Complex Hessian equations with prescribed singularity on compact Kähler manifolds (2019) (https://arxiv.org/abs/1909.02469. To appear in Annali della Scuola Normale Superiore di Pisa)

[51] Nguyen, Ngoc Cuong The complex Monge–Ampère type equation on compact Hermitian manifolds and applications, Adv. Math., Volume 286 (2016), pp. 240-285 | DOI | MR | Zbl

[52] Tosatti, Valentino Regularity of envelopes in Kähler classes, Math. Res. Lett., Volume 25 (2018) no. 1, pp. 281-289 | DOI | MR | Zbl

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

[54] Vu, Duc-Viet Complex Monge–Ampère equation for measures supported on real submanifolds, Math. Ann., Volume 372 (2018) no. 1-2, pp. 321-367 | DOI | MR | Zbl

[55] Vu, Duc-Viet Families of Monge–Ampère measures with Hölder continuous potentials, Proc. Am. Math. Soc., Volume 146 (2018) no. 10, pp. 4275-4282 | DOI | MR | Zbl

[56] Yau, Shing-Tung On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I, Commun. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 | DOI | Zbl

[57] Zeriahi, Ahmed Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions, Indiana Univ. Math. J., Volume 50 (2001) no. 1, pp. 671-703 | DOI | MR | Zbl

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