$\mathrm{CMC\text{-}1}$ surfaces in hyperbolic and de Sitter spaces with Cantor ends

TL;DR

This paper constructs new examples of constant mean curvature one surfaces with Cantor set ends in hyperbolic and de Sitter spaces, using advanced approximation techniques for holomorphic null curves.

Contribution

It introduces novel uniform approximation theorems for holomorphic null curves, enabling the construction of CMC-1 surfaces with Cantor ends in hyperbolic and de Sitter spaces.

Findings

Existence of CMC-1 surfaces with Cantor ends in hyperbolic space.

Existence of almost proper CMC-1 faces in de Sitter space.

Development of new uniform approximation theorems for holomorphic null curves.

Abstract

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 ($\mathrm{CMC\text{-}1}$) immersion into hyperbolic $3$-space $\mathbb{H}^3$. Moreover, we obtain that every bordered Riemann surface admits an almost proper $\mathrm{CMC\text{-}1}$ 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 $\mathrm{CMC\text{-}1}$ 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.