Subharmonicity of conic Mabuchi’s functional, I
Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 805-845.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3178
Classification: 32U05, 53C55, 35J35
Keywords: Mabuchi’s functional, variational method, cscK metrics
Mot clés : fonction de Mabuchi, méthode variationelle, métriques cscK

Li, Long 1

1 Department of Mathematics and Statistics McMaster University 1280 Main Street West Hamilton, ON L8S 4K1 (Canada)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2018__68_2_805_0,
     author = {Li, Long},
     title = {Subharmonicity of conic {Mabuchi{\textquoteright}s} functional, {I}},
     journal = {Annales de l'Institut Fourier},
     pages = {805--845},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {2},
     year = {2018},
     doi = {10.5802/aif.3178},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3178/}
}
TY  - JOUR
AU  - Li, Long
TI  - Subharmonicity of conic Mabuchi’s functional, I
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 805
EP  - 845
VL  - 68
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3178/
DO  - 10.5802/aif.3178
LA  - en
ID  - AIF_2018__68_2_805_0
ER  - 
%0 Journal Article
%A Li, Long
%T Subharmonicity of conic Mabuchi’s functional, I
%J Annales de l'Institut Fourier
%D 2018
%P 805-845
%V 68
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3178/
%R 10.5802/aif.3178
%G en
%F AIF_2018__68_2_805_0
Li, Long. Subharmonicity of conic Mabuchi’s functional, I. Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 805-845. doi : 10.5802/aif.3178. https://aif.centre-mersenne.org/articles/10.5802/aif.3178/

[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 | DOI | Zbl

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

[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 | DOI | Zbl

[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 | DOI | Zbl

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

[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 | DOI | Zbl

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

[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 | DOI | Zbl

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

[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 | DOI | Zbl

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

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

[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 | DOI | Zbl

Cited by Sources: