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

Abstract

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