Extremal omega-plurisubharmonic functions as envelopes of disc functionals

2009/06/04 by Benedikt Steinar Magnússon, Magnusson, Benedikt Steinar
Mathematics · #32U05 #32U40 #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.0906.0902

openalex publication_date 2009/06/04 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

For each closed, positive (1,1)-current ωon a complex manifold X and each ω-upper semicontinuous function ϕon X we associate a disc functional and prove that its envelope is equal to the supremum of all ω-plurisubharmonic functions dominated by ϕ. This is done by reducing to the case where ωhas a global potential. Then the result follows from Poletsky's theorem, which is the special case ω=0. Applications of this result include a formula for the relative extremal function of an open set in X and, in some cases, a description of the ω-polynomial hull of a set.

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