Stability of solutions to complex Monge–Ampère flows
[Stabilité des solutions de flots de Monge–Ampère complexes]
Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2819-2836.

Nous établissons un résultat de stabilité pour les équations de Monge–Ampère complexes elliptiques et paraboliques sur les variétés Kähleriennes compactes, qui s’appliquent en particulier au flot de Kähler–Ricci.

We establish a stability result for elliptic and parabolic complex Monge–Ampère equations on compact Kähler manifolds, which applies in particular to the Kähler–Ricci flow.

Publié le :
DOI : 10.5802/aif.3227
Classification : 53C44, 32W20, 58J35
Keywords: Monge–Ampère, stability, Kähler–Ricci flow
Mot clés : Monge–Ampère, stabilité, flot de Kähler–Ricci
Guedj, Vincent 1 ; Lu, Chinh H. 2 ; Zeriahi, Ahmed 1

1 Institut de Mathématiques de Toulouse Université de Toulouse, CNRS UPS IMT 118 route de Narbonne 31062 Toulouse cedex 09 (France)
2 Laboratoire de Mathématiques d’Orsay Univ. Paris-Sud, CNRS Université Paris-Saclay 91405 Orsay (France)
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     title = {Stability of solutions to complex {Monge{\textendash}Amp\`ere} flows},
     journal = {Annales de l'Institut Fourier},
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Guedj, Vincent; Lu, Chinh H.; Zeriahi, Ahmed. Stability of solutions to complex Monge–Ampère flows. Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2819-2836. doi : 10.5802/aif.3227. http://archive.numdam.org/articles/10.5802/aif.3227/

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

[2] Bedford, Eric; Taylor, Bert A. A new capacity for plurisubharmonic functions, Acta Math., Volume 149 (1982) no. 1-2, pp. 1-40 | DOI | MR

[3] Błocki, Zbigniew Uniqueness and stability for the complex Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J., Volume 52 (2003) no. 6, pp. 1697-1701 | DOI | MR

[4] 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 | MR

[5] Cegrell, Urban; Kołodziej, Sławomir; Zeriahi, Ahmed Maximal subextensions of plurisubharmonic functions, Ann. Fac. Sci. Toulouse, Math., Volume 20 (2011) no. S2, pp. 101-122 | MR

[6] Darvas, Tamás; 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

[7] Demailly, Jean-Pierre Potential Theory In Several Complex Variables, 1989 (Course of the author at the ICPAM Summer School on Complex Analysis, Nice, France, July 3–7, available at https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/nice_cimpa.pdf)

[8] Demailly, Jean-Pierre Applications of pluripotential theory to algebraic geometry, Pluripotential theory (Lecture Notes in Mathematics), Volume 2075, Springer, 2013, pp. 143-263 | DOI | MR

[9] 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 | MR

[10] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Viscosity solutions to degenerate complex Monge-Ampère equations, Commun. Pure Appl. Math., Volume 64 (2011) no. 8, pp. 1059-1094 | DOI | MR

[11] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Weak solutions to degenerate complex Monge-Ampère flows II, Adv. Math., Volume 293 (2016), pp. 37-80 | DOI | MR

[12] 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 | MR

[13] Guedj, Vincent; Zeriahi, Ahmed Degenerate complex Monge-Ampère equations, EMS Tracts in Mathematics, 26, European Mathematical Society, 2017, xxiv+472 pages | DOI | MR

[14] 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 | MR | Zbl

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

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

[17] Nguyen, Ngoc-Cuong Weak solutions to the complex Hessian equation, Jagiellonian University (Poland) (2014) (Ph. D. Thesis)

[18] 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 | MR

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