Embedding bordered Riemann surfaces in strongly pseudoconvex domains

TL;DR

This paper proves that any bordered Riemann surface can be properly embedded or immersed into strongly pseudoconvex domains in complex space, with approximation and interpolation capabilities, extending classical embedding results.

Contribution

It establishes the existence of proper holomorphic maps from bordered Riemann surfaces into strongly pseudoconvex domains that are smooth up to the boundary, with embedding and approximation properties.

Findings

Existence of proper holomorphic maps extending smoothly to the boundary

Embedding results for dimensions n≥4

Approximation and interpolation of holomorphic maps

Abstract

We show that every bordered Riemann surface, $M$, with smooth boundary $bM$ admits a proper holomorphic map $M\to \Omega$ into any bounded strongly pseudoconvex domain $\Omega$ in $\mathbb C^n$, $n>1$, extending to a smooth map $f:\overline M\to\overline \Omega$ which can be chosen an immersion if $n\ge 3$ and an embedding if $n\ge 4$. Furthermore, $f$ can be chosen to approximate a given holomorphic map $\overline M\to \Omega$ on compacts in $M$ and interpolate it at finitely many given points in $M$.