Embedding complete holomorphic discs through discrete sets

TL;DR

This paper proves that any discrete set in the unit ball of C^2 can be embedded via a complete, proper holomorphic map from the unit disc, extending the understanding of complex embeddings and their geometric properties.

Contribution

It introduces a method to embed any discrete subset of the unit ball in C^2 with a complete, proper holomorphic embedding from the unit disc, a novel construction in complex analysis.

Findings

Every discrete subset of B is contained in the image of a complete, proper holomorphic embedding.

The embedding ensures paths approaching the boundary have infinite length, demonstrating completeness.

The result advances the theory of holomorphic embeddings and their geometric properties.

Abstract

Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for any path p:[0,1)-->U such that |p(t)|-->1 as t-->1, the path t--> f(p(t)) from [0,1) to B has infinite length.