[Solutions non quadratiques de l’équation de Monge–Ampère]
We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge–Ampère equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb{C}^2$. Among these, the entire solutions defined on $\mathbb{C}^2$ induce flat Kähler metrics, as expected by a question of Calabi. In contrast, those on cylindrical domains produce a family of nowhere flat Kähler metrics. Beyond these smooth solutions, we also classify solutions that are radially symmetric in one variable, which exhibit various types of singularities. Finally, we explore analogous solutions to Donaldson’s equation motivated by a result of He.
Nous construisons de nombreuses solutions lisses, strictement plurisousharmoniques et non quadratiques de l’équation de Monge–Ampère, définies soit sur des domaines de type cylindrique, soit sur tout l’espace euclidien complexe $\mathbb{C}^2$. Parmi celles-ci, les solutions entières définies sur $\mathbb{C}^2$ induisent des métriques kählériennes plates, conformément à une question posée par Calabi. En revanche, celles définies sur des domaines cylindriques produisent une famille de métriques kählériennes nulle part plates. Au-delà de ces solutions régulières, nous classifions également les solutions radialement symétriques en une variable, qui présentent divers types de singularités. Enfin, nous examinons des solutions analogues de l’équation de Donaldson, motivés par un résultat de He.
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Keywords: Monge–Ampère equation, flat metric, radial, explicit examples, Donaldson’s equation
Yifei Pan 1 ; Yuan Zhang 1
CC-BY 4.0
Yifei Pan; Yuan Zhang. Non-quadratic solutions to the Monge–Ampère equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 2, pp. 489-516. doi: 10.5802/afst.1855
@article{AFST_2026_6_35_2_489_0,
author = {Yifei Pan and Yuan Zhang},
title = {Non-quadratic solutions to the {Monge{\textendash}Amp\`ere} equation},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {489--516},
year = {2026},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 35},
number = {2},
doi = {10.5802/afst.1855},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1855/}
}
TY - JOUR AU - Yifei Pan AU - Yuan Zhang TI - Non-quadratic solutions to the Monge–Ampère equation JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2026 SP - 489 EP - 516 VL - 35 IS - 2 PB - Université de Toulouse, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1855/ DO - 10.5802/afst.1855 LA - en ID - AFST_2026_6_35_2_489_0 ER -
%0 Journal Article %A Yifei Pan %A Yuan Zhang %T Non-quadratic solutions to the Monge–Ampère equation %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2026 %P 489-516 %V 35 %N 2 %I Université de Toulouse, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1855/ %R 10.5802/afst.1855 %G en %F AFST_2026_6_35_2_489_0
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