Properness of CMC-1 faces in de Sitter 3-space

Determine whether the existence results established for weakly complete almost proper constant mean curvature one (CMC-1) faces in de Sitter 3-space S^3_1 can be upgraded to proper CMC-1 faces. Specifically, ascertain whether (i) every bordered Riemann surface admits a weakly complete proper CMC-1 face into S^3_1, and (ii) for every compact Riemann surface M there exists a Cantor set C ⊂ M such that M \ C admits a weakly complete proper CMC-1 face into S^3_1; equivalently, decide whether the conclusions of Theorem th:bds and Theorem th:mainS hold with "almost proper" replaced by "proper."

Background

The paper develops complex analytic methods to construct CMC-1 surfaces in hyperbolic space H^3 and CMC-1 faces in de Sitter space S^3_1 using holomorphic null curves in C^2 × C*. A projection T: C^2 × C* → SL and π_S: SL → S^3_1 maps such curves to CMC-1 faces.

Two main results concerning S^3_1 are proved: Theorem th:bds shows that every bordered Riemann surface admits a weakly complete almost proper CMC-1 face into S^3_1, and Theorem th:mainS shows that for every compact Riemann surface M there exists a Cantor set C ⊂ M such that M \ C admits a weakly complete almost proper CMC-1 face into S^3_1.

Because the projection πS is not proper (SU{1,1} is non-compact), obtaining properness is technically more challenging. The authors therefore leave open whether their almost proper constructions can be strengthened to proper CMC-1 faces.