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 ⊂ M such that M C admits a proper conformal constant mean curvature one (CMC-1) immersion into hyperbolic $3$-space H³. Moreover, we obtain that every bordered Riemann surface admits an almost proper CMC-1 face into de Sitter $3$-space S₁³, and we show that on every compact Riemann surface M there is a Cantor set C ⊂ M such that M C admits an almost proper CMC-1 face into S₁³. These results follow from different uniform approximation theorems for holomorphic null curves in C² × C^* that we also establish in this paper.