Proper holomorphic embeddings of complements of large Cantor sets in $\mathbb{C}^2$

Abstract

We present a construction of a proper holomorphic embedding $f\colon \Bbb P^1\setminus C\hookrightarrow \Bbb C^2$, 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.