| In this thesis we demonstrated the existence of domains in C2 evidencing both intrinsic phenomena of Cn, n > 1, and different types of boundary smoothness. We constructed these domains by taking limits of preimages of polydiscs under a sequence of shears selected to control boundary smoothness.;Unlike the complex plane, in Cn there are simply connected domains that are biholomorphic to Cn but are proper subsets of Cn. These domains are called FatouBieberbach domains and they arise naturally in the study of complex dynamics. We showed that there exists a Fatou-Bieberbach domain in C2 with Gevrey smooth boundary.;Another interesting occurrence in C2 is the existence of simply connected proper subsets of C2 that are not biholomorphic to the unit ball nor biholomorphic to C2. Onesuch class are Short-C2 domains. We constructed Short-C2 domains with C-infinity boundary and Short-C2 domains with prescribed local Cl boundary smoothness and controlled geometry. |
