Proper Holomorphic Embeddings of complements of large Cantor sets in ${\mathbb{C}}^2$

Abstract

We present a construction of a proper holomorphic embedding $f:{\mathbb{P}}^1\setminus C\hookrightarrow {\mathbb{C}}^2$, where C is a Cantor set obtained by removing smaller and smaller vertical and horizontal strips from a square of side 2, allowing to realize it to have Lebesgue measure arbitrarily close to 4.