CMC-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 M C ⊂ M such that M C M \ C admits a proper conformal constant mean curvature one (CMC-1 CMC-1) immersion into hyperbolic 3-space H³ H 3. Moreover, we obtain that every bordered Riemann surface admits an almost proper CMC-1 CMC-1 face into de Sitter 3-space S₁³ S 1 3, and we show that on every compact Riemann surface M, there is a Cantor set C M C ⊂ M such that M C M \ C admits an almost proper CMC-1 CMC-1 face into S₁³ S 1 3. These results follow from different uniform approximation theorems for holomorphic null curves in C² C^* C 2 × C ∗ that we also establish in this paper.