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Povzetek: This paper brings several contributions to the classical Forster-Bell-Narasimhan conjecture and the Yang problem concerning the existence of proper and almost proper (hence complete) injective holomorphic immersions of open Riemann surfaces in the affine plane ${\mathbb C}^2$ satisfying interpolation and hitting conditions. We also show that in every compact Riemann surface there is a Cantor set whose complement admits a proper holomorphic embedding in ${\mathbb C}^2$. The focal point is a lemma saying the following. Given a compact bordered Riemann surface, $M$, a closed discrete subset $E$ of its interior ${\mathring M}=M\setminus bM$, a compact subset $K\subset {\mathring M}\setminus E$ without holes in $\mathring M$, and a ${\cal C}^1$ embedding $f: M\hookrightarrow \mathbb C^2$ which is holomorphic in $\mathring M$, we can approximate $f$ uniformly on $K$ by a holomorphic embedding $F: bM\hookrightarrow {\mathbb C}^2$ which maps $E\cup bM$ out of a given ball and satisfies some interpolation conditions.
Ključne besede: Riemann surfaces, complex curves, complete holomorphic embedding
Objavljeno v DiRROS: 15.07.2024; Ogledov: 19; Prenosov: 5
Celotno besedilo (579,03 KB)
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