Stein Manifolds And Holomorphic Mappings PDF Download

In the dissertation we show that every class of the first de Rham cohomology group on a Stein manifold $X$ has a representative, which is a closed holomorphic 1-form without zeros. The second set of problems in the thesis is related to embedding open Riemann surfaces properly into ${\mathbb C}^2$. In all results regarding proper holomorphic embeddings of planar domains into ${\mathbb C}^2$ that we are familiar with, the planar domain is only allowed to have finitely many boundary curves. In the dissertation we construct proper holomorphic embeddings in ${\mathbb C}^2$ for certain planar domains having infinitely many boundary curves. Finding a proper embedding for a general open Riemann surface seems very difficult. It does not appear to be easier if we omit properness. Thus, it is also interesting to relate holomorphic embeddings with proper holomorphic embeddings. Given a bordered Riemann surface $R$, embedded in ${\mathbb C}^2$, we prove that certain infinitely connected domains $D \subset R$ without isolated boundary points admit a proper holomorphic embedding into ${\mathbb C}^2$. We conclude the thesis by proving a result on approximating certain proper smooth embeddings by proper holomorphic embeddings.