High Energy plurisubharmonic classes

2025/10/18 by Vincent Guedj, Guedj, Vincent, Ahmed Zériahi +1
Mathematics · #Geometry and complex manifolds #Holomorphic and Operator Theory #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2510.16412

Abstract

Let Ω\Subset \Cⁿ be a bounded strongly pseudoconvex domain. For any concave increasing weight χ: \R^- \longrightarrow \R^- such that χ(0) = 0, we introduce and study finite energy classes \mathcal E_χ(Ω) of plurisubharmonic functions, using the Orlicz space formalism. We investigate the range of the Monge-Ampère operator on these classes, and conjecture that this should lead to an integral characterization of the image of bounded plurisubharmonic functions, an open problem since the birth of Pluripotential Theory more than forty years ago.

Citations

• The Mabuchi geometry of low energy classes
• Finite entropy vs finite energy
• Variational and non-Archimedean aspects of the Yau--Tian--Donaldson\n conjecture
• Toric Pluripotential Theory
• Open problems in pluripotential theory
• Plurisubharmonic functions with weak singularities
• The weigthed Monge-Ampère energy of quasiplurisubharmonic functions

Related

• Plurisubharmonic functions with weak singularities
• Weighted pluricomplex energy
• On some weighted energy classes of plurisubharmonic functions
• Non-pluripolar energy and the complex Monge-Ampère operator
• Subextension of plurisubharmonic functions with weak singularities