Positive Harmonic Functions that Vanish on a Subset of a Cylindrical Surface

This paper investigates positive harmonic functions on domains that are complementary to a subset of a cylindrical surface. It characterizes, both in terms of harmonic measure and of a Wiener-type criterion, those domains that admit minimal harmonic functions with exponential growth. Illustrative examples are provided. Two applications are also given. The first of these concerns minimal harmonic functions associated with an irregular boundary point, and amplifies a recent construction of Gardiner and Hansen. The second concerns the possible non-approximability of positive harmonic functions by integrable positive harmonic functions.