The Siciak–Zahariuta extremal function as the envelope of disc functionals

Abstract

Abstract. We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. This function is also known as the pluricomplex Green function with logarithmic growth or a logarithmic pole at infinity. We extend Lempert's formula for this function from the convex case to the connected case.Introduction. The Siciak-Zahariuta extremal function V X of a subset X of complex affine space C n is defined as the supremum of all entire plurisubharmonic functions u of minimal growth with u|X ≤ 0. It is also called the pluricomplex Green function of X with logarithmic growth or with a logarithmic pole at infinity (although this is a bit of a misnomer if X is not bounded). A plurisubharmonic function u on C n is said to have minimal growth (and belong to the class L) if u − log

Siciak–Zahariuta extremal functions, analytic discs and polynomial hulls

“…In [4], a similar lemma was established along the lines of Poletsky's original proof of the fundamental theorem [7]. The present authors spent a great deal of time unsuccessfully trying to give a similar proof of Lemma 2.…”

“…We denote by A X P n the set of closed analytic discs f in P n with f (0) ∈ C n and f (T) ⊂ X. In [4], we proved that if X is connected, then the Siciak-Zahariuta extremal function V X of X satisfies the disc formula…”

“…The first of our two disc formulas is analogous to the formula in [4], but has a modified functional and a larger class of discs. In [4], we showed that for every connected open subset X of C n , the Siciak-Zahariuta extremal function of X is the envelope of the disc functional J with…”

“…To prove Theorem 1, as in the proof in [4] of the analogous formula V X = E A X P n J for a connected open set X, we use our idea of a "good" auxiliary class of analytic discs, but the class is now different. The proof of the key lemma, here Lemma 2, is also different and uses a construction due to Bu and Schachermayer [1].…”

“…In previous work [4], we treated the connected case, motivated by Lempert's disc formula for the convex case ( [6], Appendix). The first of our two disc formulas is analogous to the formula in [4], but has a modified functional and a larger class of discs.…”