In this paper we study the pluripolar hulls of analytic sets. In particular, we show that hulls of graphs of analytic functions can be multiple sheeted and sheets can be separated by a set of zero dimension.

2005-03-16math.CV

The pluripolar hull of the graph of a holomorphic function with polar singularities

We study the pluripolar hull of the graph of a holomorphic function f, defined on a domain D in the complex plane outside a polar set A of D. This leads to a theorem that describes under what conditions f is nowhere extendable over A, while the graph of f over D \ A is NOT complete pluripolar.

2002-08-02math.CV

The Pluripolar Hull of a Graph and Fine Analytic Continuation

We show that if the graph of a bounded analytic function in the unit disk $\mathbb D$ is not complete pluripolar in $\mathbb C^2$ then the projection of the closure of its pluripolar hull contains a fine neighborhood of a point $p \in \partial \mathbb D$. On the other hand we show that if an analytic function $f$ in $\mathbb D$ extends to a function $\mathcal{F}$ which is defined on a fine neighborhood of a point $p \in \partial \mathbb D$ and is finely analytic at $p$ then the pluripolar hull of the graph of $f$ contains the graph of $\mathcal{F}$ over a smaller fine neighborhood of $p$. We g…

2004-05-03math.CV

Determination of the pluripolar hull of graphs of certain holomorphic functions

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

2003-10-16math.CV

Graphs that are not complete pluripolar

Let D_1 be a subdomain of D_2 in the complex plane CC. Under very mild conditions on D_2 we show that there exist holomorphic functions f, defined on D_1 with the property that $f$ is nowhere extendible across the boundary of D_1, while the graph of f over D_1 is NOT complete pluripolar in D_2 times CC. This refutes a conjecture of Levenberg, Martin and Poletsky.

2002-03-07math.CV

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Pluripolar Hulls and Fine Holomorphy

Examples by Poletsky and the author and by Zwonek show the existence nowhere extendable holomorphic functions with the property that the pluripolar hull of their graphs is much larger than the graph of the respective functions and contains multiple sheets. We will explain this phenomenon by fine analytic continuation of the function over part of a Cantor-type set involved. This gives more information on the hull, and allows for weakening and effectiveness of the conditions in the original examples.

2022-10-04math.CV