Minimal Hulls, Null Hulls, and Currents

In this chapter we initiate the study of minimal hulls and null hulls of compact sets in Euclidean spaces. These hulls are defined by the maximum principle for minimal and null plurisubharmonic functions, in analogy to the classical polynomial and plurisubharmonic hulls of complex analysis. We describe these hulls in terms of bounded sequences of conformal minimal discs or holomorphic null discs whose boundaries converge to the given compact set in measure. This extends the Poletsky-Bo-Schachermayer description of polynomial hulls to minimal and null hulls. We also show that bounded sequences of minimal and null discs give rise to minimal Green currents and positive null currents characterizing the respective hulls.