Every bordered Riemann surface is a complete proper curve in a ball

TL;DR

This paper proves that any bordered Riemann surface can be holomorphically immersed or embedded into complex Euclidean balls of dimensions 2 and 3, respectively, with completeness and properness.

Contribution

It establishes the existence of complete proper holomorphic immersions and embeddings of bordered Riemann surfaces into complex balls, extending previous results in complex analysis.

Findings

Existence of complete proper holomorphic immersions into C^2

Existence of complete proper holomorphic embeddings into C^3

Extension of classical results to bordered Riemann surfaces

Abstract

We prove that every bordered Riemann surface admits a complete proper holomorphic immersion into a ball of C^2, and a complete proper holomorphic embedding into a ball of C^3.