On the regularity problem of complex Monge–Ampere equations with conical singularities

Chen, Xiuxiong; Wang, Yuanqi

Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 969-1003.

Dans la catégorie des métriques à singularités coniques autour d’un diviseur lisse avec angle strictement compris entre $0$ et $2\pi$, on montre que les solutions faibles localement définies (au sens $C^{(1,1)}$) des équations de Kähler–Einstein sont de régularité maximale, ce qui implique que les métriques sont Höldériennes dans les coordonnées polaires singulières. Ceci montre que les métriques de Kähler–Einstein faibles construites par Guénancia-Păun, et Yao indépendamment, sont en fait des métriques de Kähler–Einstein coniques au sens fort. Le point clé est d’établir un théorème de type Liouville pour les métriques Ricci-plates au sens faible sur ${ℂ}^n$, ce qui découle d’une théorie de Calderon–Zygmund dans un contexte conique. La régularité des métriques de Kähler–Einstein coniques globalement définies avait déjà été obtenue par Guénancia-Păun par une autre méthode.

In the category of metrics with conical singularities along a smooth divisor with angle in $(0,2\pi )$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kähler–Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coordinates. This shows the weak Kähler–Einstein metrics constructed by Guenancia–Păun, and independently by Yao, are all actually strong-conical Kähler–Einstein metrics. The key step is to establish a Liouville-type theorem for weak-conical Kähler–Ricci flat metrics defined over ${ℂ}^n$, which depends on a Calderon–Zygmund theory in the conical setting. The regularity of globally defined weak-conical Kähler–Einstein metrics is already proved by Guenancia–Paun using a different method.

Classification : 35J75
Keywords: complex Monge–Ampère equations, conical singularity
Mot clés : équations complexes de Monge–Ampère, singularité conique

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