Embedded complex curves in the affine plane

This paper brings several contributions to the classical Forster–Bell–Narasimhan conjecture and the Yang problem concerning the existence of proper, almost proper, and complete injective holomorphic immersions of open Riemann surfaces in the affine plane \(\mathbb {C}^2\) satisfying interpolation and hitting conditions. We also show that every compact Riemann surface contains a Cantor set whose complement admits a proper holomorphic embedding in \(\mathbb {C}^2\), and every connected domain in \(\mathbb {C}^2\) admits complete, everywhere dense, injectively immersed complex discs. The focal point of the paper is a lemma saying for every compact bordered Riemann surface, M, closed discrete subset E of \(\mathring{M}=M\setminus bM\), and compact subset \(K\subset \mathring{M}\setminus E\) without holes in \(\mathring{M}\), any \(\mathscr {C}^1\) embedding \(f:M\hookrightarrow \mathbb {C}^2\) which is holomorphic in \(\mathring{M}\) can be approximated uniformly on K by holomorphic embeddings \(F:M\hookrightarrow \mathbb {C}^2\) which map \(E\cup bM\) out of a given ball and satisfy some interpolation conditions.