Plurisubharmonic subextensions as envelopes of disc functionals

TL;DR

This paper establishes a disc formula for the largest plurisubharmonic subextension in Stein manifolds, introduces an equivalence relation on analytic discs, and generalizes Kiselman's minimum principle using these concepts.

Contribution

It provides a new disc formula for plurisubharmonic subextensions and links it to a novel equivalence relation on analytic discs, extending Kiselman's minimum principle.

Findings

Derived a disc formula for subextensions in Stein manifolds

Introduced an equivalence relation on analytic discs with boundary in W

Generalized Kiselman's minimum principle

Abstract

We prove a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain $W$ in a Stein manifold to a larger domain $X$ under suitable conditions on $W$ and $X$. We introduce a related equivalence relation on the space of analytic discs in $X$ with boundary in $W$. The quotient, if it is Hausdorff, is a complex manifold with a local biholomorphism to $X$. We use our disc formula to generalise Kiselman's minimum principle. We show that his infimum function is an example of a plurisubharmonic subextension.