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Embedding subsets of tori Properly into ℂ 2

Erlend Fornæss Wold — 2007

Annales de l’institut Fourier

Let 𝕋 be a complex one-dimensional torus. We prove that all subsets of 𝕋 with finitely many boundary components (none of them being points) embed properly into ℂ 2 . We also show that the algebras of analytic functions on certain countably connected subsets of closed Riemann surfaces are doubly generated.

Riemann surfaces in Stein manifolds with the Density property

Rafael B. Andrist;  Erlend Fornæss Wold — 2014

Annales de l’institut Fourier

We show that any open Riemann surface can be properly immersed in any Stein manifold with the (Volume) Density property and of dimension at least 2. If the dimension is at least 3, we can actually choose this immersion to be an embedding. As an application, we show that Stein manifolds with the (Volume) Density property and of dimension at least 3, are characterized among all other complex manifolds by their semigroup of holomorphic endomorphisms.

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