Bryant showed that in a space of constant sectional curvature - c2 (for c ∈ R ), there is a representation for surfaces which have constant mean curvature equal to c, abbreviated as CMC c, which is analogous to the Weierstrass representation in R3 . For c > 0, by rescaling the metric, we can limit ourselves to examining surfaces with CMC 1, immersed in hyperbolic 3-space H3 .
Many examples of minimal surfaces in R3 have found their analogs (cousins, in the terminology of Bryant) in the new theory. However there is one minimal surface, a thrice-punctured torus discovered by Costa, whose analog has resisted discovery.
In this work we build on the results of Bryant, Umehara and Yamada to show that for a particular thrice punctured torus M, there exists a countable family of complete, finite total curvature, CMC 1, singly branched immersions into H3 having regular ends. We also show that this family is the unique family of such immersions whose Weierstrass data has a certain form. The proof of the existence and uniqueness of this family of singly branched Costa cousins is divided into three parts. In part I we prove that out of a given set of Weierstrass data there is only one real parameter family of candidates which can yield a complete, finite total curvature, possibly singular, CMC 1 immersion. In this case, Bryant's Weierstrass, Representation theorem applies to give us the existence of a multi-valued singular CMC 1 immersion of M into H3. In part II we show that such a singular immersion is well-defined across the handle generators by showing that it can be written as a function of the Weierstrass ℘ -function on M. Finally, in part III, for a countable subset of the real parameter family, we show that such a singular immersion is well-defined in a neighborhood of each of the three ends. From the construction we can conclude that the immersion is a complete, finite total curvature, having regular ends, and having one singularity, which is a branch point.