\(\text {CMC-1}\) Surfaces in Hyperbolic and de Sitter Spaces with Cantor Ends

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