[La méthode de continuité de Ricci pour lʼéquation de Monge–Ampère complexe, avec des applications aux métriques de Kähler–Einstein conique le long dʼarêtes]
Dans cette Note nous introduisons une nouvelle méthode de continuité et estimée a priori, pour lʼéquation de Monge–Ampère complexe dégénérée. Nous présentons également quelques applications de cette méthode à lʼexistence de métriques de Kähler–Einstein ayant une structure conique le long dʼarêtes, confirm des conjectures de Tian et de Donaldson.
In this Note we present a new continuity method and a priori estimate for the degenerate complex Monge–Ampère equation. We then describe some applications of this method to the existence of Kähler–Einstein edge metrics, as conjectured by Tian and Donaldson.
Accepté le :
Publié le :
@article{CRMATH_2012__350_13-14_693_0,
author = {Mazzeo, Rafe and Rubinstein, Yanir A.},
title = {The {Ricci} continuity method for the complex {Monge{\textendash}Amp\`ere} equation, with applications to {K\"ahler{\textendash}Einstein} edge metrics},
journal = {Comptes Rendus. Math\'ematique},
pages = {693--697},
publisher = {Elsevier},
volume = {350},
number = {13-14},
year = {2012},
doi = {10.1016/j.crma.2012.07.001},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.07.001/}
}
TY - JOUR AU - Mazzeo, Rafe AU - Rubinstein, Yanir A. TI - The Ricci continuity method for the complex Monge–Ampère equation, with applications to Kähler–Einstein edge metrics JO - Comptes Rendus. Mathématique PY - 2012 SP - 693 EP - 697 VL - 350 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.07.001/ DO - 10.1016/j.crma.2012.07.001 LA - en ID - CRMATH_2012__350_13-14_693_0 ER -
%0 Journal Article %A Mazzeo, Rafe %A Rubinstein, Yanir A. %T The Ricci continuity method for the complex Monge–Ampère equation, with applications to Kähler–Einstein edge metrics %J Comptes Rendus. Mathématique %D 2012 %P 693-697 %V 350 %N 13-14 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.07.001/ %R 10.1016/j.crma.2012.07.001 %G en %F CRMATH_2012__350_13-14_693_0
Mazzeo, Rafe; Rubinstein, Yanir A. The Ricci continuity method for the complex Monge–Ampère equation, with applications to Kähler–Einstein edge metrics. Comptes Rendus. Mathématique, Tome 350 (2012) no. 13-14, pp. 693-697. doi : 10.1016/j.crma.2012.07.001. http://www.numdam.org/articles/10.1016/j.crma.2012.07.001/
[1] Équations du type Monge–Ampère sur les variétés kählériennes compactes, Bull. Sci. Math., Volume 102 (1978), pp. 63-95
[2] Réduction du cas positif de lʼéquation de Monge–Ampère sur les variétés kählériennes compactes à la démonstration dʼune inégalité, J. Funct. Anal., Volume 57 (1984), pp. 143-153
[3] A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics (preprint) | arXiv
[4] Ricci flat Kähler metrics with edge singularities (preprint) | arXiv
[5] On Kähler manifolds with vanishing canonical class (Fox, R.H., ed.), Algebraic Geometry and Topology. A Symposium in Honor of S. Lefschetz, Princeton University Press, 1957, pp. 78-89
[6] Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields (preprint) | arXiv
[7] Discussion of the Kähler–Einstein problem http://www2.imperial.ac.uk/~skdona/KENOTES.PDF (2009, preprint, available at)
[8] Kähler metrics with cone singularities along a divisor (preprint) | arXiv
[9] Kähler–Einstein metrics with edge singularities (with an appendix by C. Li and Y.A. Rubinstein), preprint | arXiv
[10] K-energy maps integrating Futaki invariants, Tôhoku Math. J., Volume 38 (1986), pp. 575-593
[11] Elliptic theory of differential edge operators, I, Comm. PDE, Volume 16 (1991), pp. 1616-1664
[12] R. Mazzeo, Y.A. Rubinstein, Kähler–Einstein metrics with crossing edge singularities, in preparation.
[13] Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone 3-manifolds, 2010 (preprint) | arXiv
[14] Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math., Volume 218 (2008), pp. 1526-1565
[15] Kähler–Einstein metrics on algebraic manifolds, Cetraro, 1994 (Lecture Notes in Math.), Volume vol. 1646 (1996), pp. 143-185
[16] Kähler–Einstein metrics with positive scalar curvature, Inv. Math., Volume 130 (1997), pp. 1-37
[17] Kähler–Einstein metrics of negative Ricci curvature on general quasiprojective manifolds, Comm. Anal. Geom., Volume 16 (2008), pp. 395-435
[18] On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math., Volume 31 (1978), pp. 339-411
Cité par Sources :
