A properly embedded holomorphic disc in the ball with finite area and dense boundary curve

TL;DR

This paper constructs a properly embedded holomorphic disc in the unit ball of complex two-space that has finite area and a boundary curve dense in the sphere, revealing new geometric properties of holomorphic embeddings.

Contribution

It introduces a novel example of a holomorphic disc with finite area and a dense boundary curve in the ball, combining properties previously thought incompatible.

Findings

Holomorphic disc with finite area constructed

Boundary curve is dense in the sphere

Disc is the zero set of a bounded holomorphic function

Abstract

In this paper we construct a properly embedded holomorphic disc in the unit ball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on $\mathbb{B}^2$; on the other hand, its boundary curve is everywhere dense in the sphere $b\mathbb{B}^2$.