Stability of solutions to complex Monge–Ampère flows
Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2819-2836.

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.

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.

Published online:
DOI: 10.5802/aif.3227
Classification: 53C44,  32W20,  58J35
Keywords: Monge–Ampère, stability, Kähler–Ricci flow
@article{AIF_2018__68_7_2819_0,
     author = {Guedj, Vincent and Lu, Chinh H. and Zeriahi, Ahmed},
     title = {Stability of solutions to complex {Monge{\textendash}Amp\`ere} flows},
     journal = {Annales de l'Institut Fourier},
     pages = {2819--2836},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {7},
     year = {2018},
     doi = {10.5802/aif.3227},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3227/}
}
TY  - JOUR
TI  - Stability of solutions to complex Monge–Ampère flows
JO  - Annales de l'Institut Fourier
PY  - 2018
DA  - 2018///
SP  - 2819
EP  - 2836
VL  - 68
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3227/
UR  - https://doi.org/10.5802/aif.3227
DO  - 10.5802/aif.3227
LA  - en
ID  - AIF_2018__68_7_2819_0
ER  - 
%0 Journal Article
%T Stability of solutions to complex Monge–Ampère flows
%J Annales de l'Institut Fourier
%D 2018
%P 2819-2836
%V 68
%N 7
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3227
%R 10.5802/aif.3227
%G en
%F AIF_2018__68_7_2819_0
Guedj, Vincent; Lu, Chinh H.; Zeriahi, Ahmed. Stability of solutions to complex Monge–Ampère flows. Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2819-2836. doi : 10.5802/aif.3227. https://aif.centre-mersenne.org/articles/10.5802/aif.3227/

[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 (Lecture Notes in Mathematics) Tome 2075, Springer, 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, EMS Tracts in Mathematics, Tome 26, European Mathematical Society, 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 (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

Cited by Sources: