Stability of solutions to complex Monge–Ampère flows  [ Stabilité des solutions de flots de Monge–Ampère complexes ]
Annales de l'Institut Fourier, à paraître, 18 p.

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 : 2019-03-08
Classification:  53C44,  32W20,  58J35
Mots clés: Monge–Ampère, stabilité, flot de Kähler–Ricci
@unpublished{AIF_0__0_0_A35_0,
     author = {Guedj, Vincent and Lu, Chinh H. and Zeriahi, Ahmed},
     title = {Stability of solutions to complex Monge--Amp\`ere flows},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Guedj, Vincent; Lu, Chinh H.; Zeriahi, Ahmed. Stability of solutions to complex Monge–Ampère flows. Annales de l'Institut Fourier, à paraître, 18 p.

[1] Bedford, Eric; Taylor, Bert A. The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., Tome 37 (1976) no. 1, pp. 1-44 | Article | MR 0445006 | Zbl 0315.31007

[2] Bedford, Eric; Taylor, Bert A. A new capacity for plurisubharmonic functions, Acta Math., Tome 149 (1982) no. 1-2, pp. 1-40 | Article | MR 674165

[3] Błocki, Zbigniew Uniqueness and stability for the complex Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J., Tome 52 (2003) no. 6, pp. 1697-1701 | Article | MR 2021054

[4] Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Monge-Ampère equations in big cohomology classes, Acta Math., Tome 205 (2010) no. 2, pp. 199-262 | Article | MR 2746347

[5] Cegrell, Urban; Kołodziej, Sławomir; Zeriahi, Ahmed Maximal subextensions of plurisubharmonic functions, Ann. Fac. Sci. Toulouse, Math., Tome 20 (2011) no. S2, pp. 101-122 | MR 2858169

[6] Darvas, Tamás; Di Nezza, Eleonora; Lu, Chinh H. On the singularity type of full mass currents in big cohomology classes, Compos. Math., Tome 154 (2018) no. 2, pp. 380-409 | Article | MR 3738831 | Zbl 1398.32042

[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, Springer (Lecture Notes in Mathematics) Tome 2075 (2013), pp. 143-263 | Article | MR 3089070

[9] Dinew, Sławomir; Zhang, Zhou On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds, Adv. Math., Tome 225 (2010) no. 1, pp. 367-388 | Article | MR 2669357

[10] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Viscosity solutions to degenerate complex Monge-Ampère equations, Commun. Pure Appl. Math., Tome 64 (2011) no. 8, pp. 1059-1094 | Article | MR 2839271

[11] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Weak solutions to degenerate complex Monge-Ampère flows II, Adv. Math., Tome 293 (2016), pp. 37-80 | Article | MR 3474319

[12] Guedj, Vincent; Zeriahi, Ahmed Stability of solutions to complex Monge-Ampère equations in big cohomology classes, Math. Res. Lett., Tome 19 (2012) no. 5, pp. 1025-1042 | Article | MR 3039828

[13] Guedj, Vincent; Zeriahi, Ahmed Degenerate complex Monge-Ampère equations, European Mathematical Society, EMS Tracts in Mathematics, Tome 26 (2017), xxiv+472 pages | Article | MR 3617346

[14] Kołodziej, Sławomir Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge-Ampère operator, Ann. Pol. Math., Tome 65 (1996) no. 1, pp. 11-21 | Article | MR 1414748 | Zbl 0878.32014

[15] Kołodziej, Sławomir The complex Monge-Ampère equation, Acta Math., Tome 180 (1998) no. 1, pp. 69-117 | Article | MR 1618325

[16] Kołodziej, Sławomir The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J., Tome 52 (2003) no. 3, pp. 667-686 | Article | MR 1986892 | Zbl 1039.32050

[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., Tome 31 (1978) no. 3, pp. 339-411 | Article | MR 480350