Asympotitcs for Some Singular Monge-Ampère Equations

2024/10/19 by Nicholas McCleerey, McCleerey, Nicholas
Mathematics · #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Primary: 32U05 #Secondary: 35J60

paper · pdf · doi:10.48550/arxiv.2410.15202

openalex publication_date 2024/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a psh function φ\inE(Ω) and a smooth, bounded θ≥ 0, it is known that one can solve the Monge-Ampère equation MA(φ_θ)=θⁿMA(φ), with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czyż--Hiep. Under some natural conditions, we show that φ_θ is comparable to θφ on much of Ω; especially, it is bounded on the interior of \θ= 0\. Our results also apply to complex Hessian equations, and can be used to produce interesting Green's functions.

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