$\mathrm{CMC}\text{-}1$ surfaces in hyperbolic and de Sitter spaces with Cantor ends

Abstract

We prove that on every compact Riemann surface $M$ there is a Cantor set $C\subset M$ such that $M\setminus C$ admits a proper conformal constant mean curvature one ($\mathrm{CMC}\text{-}1$) immersion into hyperbolic $3$-space ${\mathbb{H}}^3$. Moreover, we obtain that every bordered Riemann surface admits an almost proper $\mathrm{CMC}\text{-}1$ face into de Sitter $3$-space ${\mathbb{S}}_1^3$, and we show that on every compact Riemann surface $M$ there is a Cantor set $C\subset M$ such that $M\setminus C$ admits an almost proper $\mathrm{CMC}\text{-}1$ face into ${\mathbb{S}}_1^3$. These results follow from different uniform approximation theorems for holomorphic null curves in ${\mathbb{C}}^2\times {\mathbb{C}}^{\ast }$ that we also establish in this paper.