Non-linearity of the pluricomplex Green function

Abstract

Abstract. We consider the pluricomplex Green function with multiple poles as introduced by Lelong. We give a partial solution to a question concerning the set where the multipole Green function coincides with the sum of the corresponding single pole Green functions.

Coman conjecture for the bidisc

“…Recall that l D (z; P ; ν) = min{l D (z; A; ν |A ) : ∅ = A ⊂ P } (see [14] for arbitrary D or [19] for D convex). The last equality will be of interest for us since in the case of taut domains (convex and bounded domains are taut) the infimum in the definition of l D (z; P ; ν) will be attained by some analytic disc defining l D (z; A; ν |A ) for some ∅ = A ⊂ P .…”

“…. , N.Recall that l D (z; P ; ν) = min{l D (z; A; ν |A ) : ∅ = A ⊂ P } (see [14] for arbitrary D or [19] for D convex). The last equality will be of interest for us since in the case of taut domains (convex and bounded domains are taut) the infimum in the definition of l D (z; P ; ν) will be attained by some analytic disc defining l D (z; A; ν |A ) for some ∅ = A ⊂ P .The function l D (•; P ; ν) is called the Lempert function with the poles at P and with the weight function ν (or weights ν j ).Analoguously we define the pluricomplex Green function g D (z; P ; ν) with the poles at P and the weight function ν as the supremum of 2010 Mathematics Subject Classification.…”

Convergence and multiplicities for the Lempert function

“…Then L S (z) = min S ′ ⊂S ℓ S ′ (z), where ℓ S is defined in (1.1). And in fact min S ′ ⊂S ℓ S ′ (z) = ℓ S (z) [10] (see also [17], [18] for the case when the domain Ω is convex).…”

“…The definition of a generalized Lempert function chosen in [16] had some drawbacks -essentially, it was not monotonic with respect to its system of poles (in an appropriate sense) [16,Proposition 4.3] and did not pass to the limit in some very simple situations [15,Theorem 6.3]. We recall that monotonicity holds when no multiplicities are present, see [18] and [16,Proposition 3.1] for the convex case, and the more recent [10] for arbitrary domains and weighted Lempert functions, or more generally when a subset of the original set of poles is considered with the same generalized local indicators.…”

The multipole Lempert function is monotone under inclusion of pole sets

“…where the infimum is taken over all subsets (λ j ) l j=1 of D (in this paper, D is the open unit disc in C) for which there is an analytic disc ϕ ∈ O(D, D) with ϕ(0) = z and ϕ(λ j ) = a j for all j. Here we call l D (p, •) the Lempert function with p-weighted poles at A [8,9]; see also [5], where this function is called the Coman function for p.…”