One of the biggest open problems in Complex Geometry is whether every open Riemann Surface admits a proper holomorphic embedding into the 2-dimensional euclidean complex space, as this provides a good representation of it.
We constructed examples of this representation for certain Riemann Surfaces that were thought to be good candidates for a counterexample, namely complements of large Cantor sets inside the Riemann sphere. A previous paper provided the desired representation for the same object; here we pointed out that from that construction it turns out that the resulting Cantor set is actually very thin.
Another important result contained in the thesis is the simultaneous construction of the above representation for a whole family of domains inside the Riemann sphere.
Holomorphic embeddings (non-proper) are studied in terms of approximation as well: we give sufficient conditions on domains inside euclidean complex spaces to approximate such mappings on the given domains with similar mappings with dense images. We finally extend this result to more general settings, presenting three further theorems.