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 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 S31, 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 S31. These results follow from different uniform approximation theorems for holomorphic null curves in C2 × C* that we also establish in this paper.