Continuity of envelopes of unbounded plurisubharmonic functions

2021/09/28 by Mårten Nilsson, Nilsson, Mårten · 2 citations
Mathematics · #31C05 #31C10 #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Primary 32U05 #Secondary 32U15

paper · pdf · doi:10.48550/arxiv.2109.13625

openalex publication_date 2021/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function ϕ such that ϕ^*=ϕ_* on the closure of the domain. For ϕ satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron-Bremermann envelope P ϕ is guaranteed to be continuous. This result is a generalization of a classic result in pluripotential theory due to J. B. Walsh. As an application, we show that the complex Monge--Ampère equation (dd^cu)ⁿ = μ being uniquely solvable for continuous boundary data implies that it is also uniquely solvable for a class of boundary values unbounded from above.

Cited by

• Restricted Perron envelopes and quasibounded functions
• Dirichlet problem for plurisubharmonic functions on bounded quasi B-regular domains in ℂⁿ

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