Polynomial Hulls and Envelopes of Holomorphy

Abstract: The notion of polynomial hulls of compact subsets of complex Euclidean space plays a crucial role for approximation of holomorphic functions by polynomials - a topic which has many applications. Despite an abstract characterization of polynomial hulls in terms of currents, given by Duval and Sibony, it is often very difficult even for special classes of sets to decide whether the polynomial hull is trivial (i.e. coincides with the set) or not.An interesting observation is that Oka's description of polynomial hulls using continuous families of analytic varieties links this topic to the so called "continuity principle", the principle of compulsory analytic continuation of functions of several complex variables, which is a foundation of the notion of envelopes of holomorphy.Our first result states that there are Cantor sets in the unit sphere in complex Euclidean space of dimension at least three the polynomial hull of which contains interior points. (The case of dimension two was treated earlier by B. Jöricke.) Our result may be contrasted to a fact concerning analytic continuation: Cantor sets are removable, i.e. the envelope of holomorphy of the complement of any Cantor set in the unit sphere in complex Euclidean space of dimension at least three coincides with the whole unit ball.Our second result is the construction of an open connected subset of the unit sphere in three-dimensional complex space, such that the envelope of holomorphy of this set has infinitely many sheets. On the other hand it is known that the envelope of holomorphy of an open subset of a strictly pseudoconvex boundary in two-dimensional space is always single-sheeted.Our third result is a general form of the continuity principle which emerged from the correction of a respective erroneous result in the literature.