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

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