We show the existence of constant mean curvature-one surfaces in both hyperbolic 3-space and de Sitter 3-space with two complete embedded ends and any positive genus up to genus twenty. We also find another such family of surfaces in de Sitter 3-space, but with a different non-embedded end behavior-c.
- Constant mean curvature-one surface
- De sitter 3-space
- Higher-genus surface
- Hyperbolic 3-space
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