ON LINEARLY CONVEX HARTOGS REGIONS IN C2, WITH FRACTAL STRUCTURE

Abstract

In the 1970s, it was proved that a bounded linearly convex domain with a smooth boundary in Cn is homeomorphic to an open ball. If the boundary of a bounded linearly convex domain in Cn is not smooth, then the domain may have different topological types. Only for n=2 complex plane projection a1z1 + . . . + anzn + c = 0 to the Hartogs (Hartogs) diagram in Cn with symmetry plane zn = 0 has a simple geometric form: it is a circular cone with vertex on the plane z2 = 0. This fact allows one to construct linearly convex Hartogs domains in C2 with symmetry plane z2 = 0, whose projection onto the Hartogs diagram has a fractal structure.