Determination of the pluripolar hull of graphs of certain holomorphic functions

Abstract

Let A be a closed polar subset of a domain D in C. We give a complete description of the pluripolar hull Γ * D×C of the graph Γ of a holomorphic function definedon D \ A. To achieve this, we prove for pluriharmonic measure certain semi-continuity properties and a localization principle.

Pluripolar Hulls and Complete Pluripolar Sets

“…The next result, Proposition 2.7, is an attempt to describe pluripolar hulls by currents. Section 3 begins with Theorem 3.1 which states that if F is a complete pluripolar subset of the complement of a closed complete pluripolar E in a pseudoconvex domain D, then E ∪ F is complete pluripolar in D. Using this result and some ideas from previous work of Edigarian and Wiegerinck, we are able to give in Theorem 3.3, a partial generalization of the main result in [9], where holomorphic graph is replaced by complex subvariety. Next, in Theorem 3.4 we show, roughly speaking, that the pluripolar hull of a pluripolar set lying outside a smooth complex hypersurface can not contain entirely the complex hypersurface.…”

“…However, a complete description of ( f ) * D is still missing. Turning to the present work, in Section 2, we collect preliminary facts about pluripolar sets which include known results taken from [12] and [9]. We construct an example of a closed pluripolar subset E in a domain D such that E ∩ D is completely pluripolar in D for every subdomain D of D which belongs compactly to D, but E is not completely pluripolar in D. It should be remarked that such an example can not exist if D is pseudoconvex.…”

“…The proof uses properties of the complex Monge-Ampère operator in an essential way (e.g., a slicing formula for currents due to Bedford and Taylor and a generalized Stoke formula due to Cegrell). It would be of interest to have a more elementary proof based on the tools developed in [8,12] and [9]. The last main result of the paper focuses on pluripolar hulls of holomorphic graphs in high dimension.…”

“…Given a closed complete pluripolar subset E of a domain D ⊂ C n and a complete pluripolar subset F ⊂ D\E, we study the pluripolar hull F * D . Then the knowledge on F * D will enable us to decide whether F is complete pluripolar in the domain D. We should mention that this problem, in the case D = D × C, E = E × C, where E is a closed polar subset of an open set D ⊂ C and F is the graph of a holomorphic function f on D \E , has been completely solved in the work of Edigarian and Wiegerinck (see [8] and [9]). As a first step, they show in Theorem 2.1 of [9] that F * D is a subset of E ∪ F. Then necessary and sufficient conditions on D , E and f are given so that F is complete pluripolar in D. Moreover, they show that for any point z 0 ∈ E the set ({z 0 } × C) ∩ F * D consists of at most one point.…”

Graphs with multiple sheeted pluripolar hulls

“…Edigarian and the second author showed that if D = C \ K, with K a polar compact set in C, and if f is not extendible over K, then (Γ f ) * C 2 ∩ D × C = Γ f ; see [3], and [4] for the fact that also over K the hull is at most single sheeted.…”