Attaching handles to Bryant surfaces

Abstract

We prove the existence of complete, embedded, constant mean curvature 1 surfaces in 3 dimensional hyperbolic space when g, the genus of the surface, and n, the number of ends of the surface, satisfy either g=0 and $n\geq 1$ or $g \geq 1$ and $2n\geq g+5$. The surfaces are all regular points of their corresponding moduli space.