Pluripolar Hulls and Fine Holomorphy

TL;DR

This paper explores the phenomenon where pluripolar hulls of certain holomorphic functions' graphs are significantly larger than the graphs themselves, using fine analytic continuation over Cantor-type sets to explain and refine previous examples.

Contribution

It introduces a method of fine analytic continuation to better understand and characterize pluripolar hulls of holomorphic functions' graphs, improving upon prior examples.

Findings

Pluripolar hulls can be much larger than the original graphs.

Fine analytic continuation explains the hull's structure.

Conditions for hull enlargement can be weakened and made more effective.

Abstract

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.