Extremal omega-plurisubharmonic functions as envelopes of disc functionals
Complex Variables
2010-04-13 v2
Abstract
For each closed, positive (1,1)-current \omega on a complex manifold X and each \omega-upper semicontinuous function \phi on X we associate a disc functional and prove that its envelope is equal to the supremum of all \omega-plurisubharmonic functions dominated by \phi. This is done by reducing to the case where \omega has a global potential. Then the result follows from Poletsky's theorem, which is the special case \omega=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 \omega-polynomial hull of a set.
Cite
@article{arxiv.0906.0902,
title = {Extremal omega-plurisubharmonic functions as envelopes of disc functionals},
author = {Benedikt Steinar Magnusson},
journal= {arXiv preprint arXiv:0906.0902},
year = {2010}
}
Comments
15 pages. Improved notation for the extension of $u+\psi$ over the singular set and refined Definition 2.1 of $\omega$-usc functions.