Non-quadratic solutions to the Monge–Ampère equation
Abstract
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 ℂ 2 . Among these, the entire solutions defined on 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.
Fields of Study & Subjects
Geometry and complex manifolds
Geometric Analysis and Curvature Flows
Publication details
- Journal / source
- Annales de la faculté des sciences de Toulouse Mathématiques
- Volume
- 35
- Issue
- 2
- Pages
- 489–516
- Publication date
- 12 June 2026
- Language
- EN