Subharmonicity of conic Mabuchi’s functional, I  [ Subharmonicité de la fonction de Mabuchi conique, I ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 805-845.

Le but de cet article est de démontrer la convexité de la fonctionnelle de Mabuchi le long d’une géodésique dans le cadre conique. Nous considérons d’abord les métriques de Kähler de courbure scalaire constante (cscK) et ensuite nous introduisons la fonctionnelle de Mabuchi de sorte que les métriques coniques cscK soient ses points critiques. Par la suite nous démontrons le résultat principal.

The purpose of this paper is to prove the convexity of Mabuchi’s functional along a geodesic in the conic setting. We first establish a scheme to study conic constant scalar curvature Kähler (cscK) metrics, and then the conic Mabuchi functional is introduced in such a way that conic cscK metrics are its critical points. Finally we prove that the conic Mabuchi functional is convex and continuous along a conic geodesic.

Reçu le : 2016-03-28
Révisé le : 2016-06-17
Accepté le : 2016-08-11
Publié le : 2018-04-17
DOI : https://doi.org/10.5802/aif.3178
Classification : 32U05,  53C55,  35J35
Mots clés : fonction de Mabuchi, méthode variationelle, métriques cscK
@article{AIF_2018__68_2_805_0,
     author = {Li, Long},
     title = {Subharmonicity of conic Mabuchi's functional, I},
     journal = {Annales de l'Institut Fourier},
     pages = {805--845},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {2},
     year = {2018},
     doi = {10.5802/aif.3178},
     language = {en},
     url = {archive.numdam.org/item/AIF_2018__68_2_805_0/}
}
Li, Long. Subharmonicity of conic Mabuchi’s functional, I. Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 805-845. doi : 10.5802/aif.3178. http://archive.numdam.org/item/AIF_2018__68_2_805_0/

[1] Berman, Robert J.; Berndtsson, Bo Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics, J. Am. Math. Soc., Volume 30 (2017) no. 4, pp. 1165-1196 | Article | Zbl 06750377

[2] Blocki, Zbigniew; Kolodziej, Slawomir On regularization of plurisubharmonic functions on manifolds, Proc. Am. Math. Soc., Volume 135 (2007) no. 7, pp. 2089-2093 | Article | Zbl 1116.32024

[3] 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 | Article | Zbl 1213.32025

[4] Calamai, Simone; Zheng, Kai Geodesics in the space of Kähler cone metrics I, Am. J. Math., Volume 137 (2015) no. 5, pp. 1149-1208 | Article | Zbl 1334.58006

[5] Chen, Xiuxiong The space of Kähler metrics, J. Differ. Geom., Volume 56 (2000) no. 2, pp. 189-234 | Article | Zbl 1041.58003

[6] Chen, Xiuxiong; Li, Long; Paun, Mihai Approximation of weak geodesics and subharmonicity of Mabuchi energy (2014) (https://arxiv.org/abs/1409.7896)

[7] Chen, Xiuxiong; Wang, Yuanqi On the regularity problem of complex Monge-Ampere equations with conical singularities (2014) (https://arxiv.org/abs/1405.1021)

[8] Demailly, Jean-Pierre Complex analytic and differential geometry (1997) (https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)

[9] Donaldson, Simon Kirwan Kähler metrics with cone singularities along a divisor, Essays in mathematics and its applications. In honor of Stephen Smale’s 80th birthday, Springer, 2012, pp. 49-79 | Article | Zbl 1326.32039

[10] Guedj, Vincent; Zeriahi, Ahmed The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal., Volume 250 (2007) no. 2, pp. 442-482 | Article | Zbl 1143.32022

[11] Guenancia, Henri; Paun, Mihai Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors, J. Differ. Geom., Volume 103 (2016) no. 1, pp. 15-57 | Article | Zbl 1344.53053

[12] Kołodziej, Sławomir The complex Monge-Ampère equation, Acta Math., Volume 180 (1998) no. 1, pp. 69-117 | Article | Zbl 0913.35043

[13] 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 | Article | Zbl 1149.32018

[14] Mabuchi, Toshiki K-energy maps integrating Futaki invariants, Tohoku Math. J., Volume 38 (1986) no. 1-2, pp. 575-593 | Article | Zbl 0619.53040

[15] Mabuchi, Toshiki Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math., Volume 24 (1987), pp. 227-252 | Zbl 0645.53038

[16] Păun, Mihai Relative adjoint transcendental classes and Albanese maps of compact Kaehler manifolds with nef Ricci curvature (2012) (https://arxiv.org/abs/1209.2195)

[17] Yau, Shing-Tung On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampere equation, I, Commun. Pure Appl. Math., Volume 31 (1978), pp. 339-411 | Article | Zbl 0369.53059